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2026-01-01
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<p>Last updated on<strong>September 15, 2025</strong></p>
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<p>Last updated on<strong>September 15, 2025</strong></p>
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<p>We use the derivative of e^-7x, which is -7e^-7x, to understand how the exponential function changes in response to a slight change in x. Derivatives are crucial in calculating rates of change in real-life situations. We will now discuss the derivative of e^-7x in detail.</p>
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<p>We use the derivative of e^-7x, which is -7e^-7x, to understand how the exponential function changes in response to a slight change in x. Derivatives are crucial in calculating rates of change in real-life situations. We will now discuss the derivative of e^-7x in detail.</p>
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<h2>What is the Derivative of e^-7x?</h2>
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<h2>What is the Derivative of e^-7x?</h2>
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<p>We now understand the derivative<a>of</a>e^-7x. It is commonly represented as d/dx (e^-7x) or (e^-7x)', and its value is -7e^-7x. The<a>function</a>e^-7x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:</p>
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<p>We now understand the derivative<a>of</a>e^-7x. It is commonly represented as d/dx (e^-7x) or (e^-7x)', and its value is -7e^-7x. The<a>function</a>e^-7x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:</p>
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<p><strong>Exponential Function:</strong>e^-7x is an exponential function with a<a>base</a>of e.</p>
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<p><strong>Exponential Function:</strong>e^-7x is an exponential function with a<a>base</a>of e.</p>
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<p><strong>Chain Rule:</strong>Rule for differentiating composite functions.</p>
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<p><strong>Chain Rule:</strong>Rule for differentiating composite functions.</p>
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<p><strong>Constant Multiple Rule:</strong>Enables differentiation of functions multiplied by a<a>constant</a>.</p>
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<p><strong>Constant Multiple Rule:</strong>Enables differentiation of functions multiplied by a<a>constant</a>.</p>
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<h2>Derivative of e^-7x Formula</h2>
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<h2>Derivative of e^-7x Formula</h2>
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<p>The derivative of e^-7x can be denoted as d/dx (e^-7x) or (e^-7x)'.</p>
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<p>The derivative of e^-7x can be denoted as d/dx (e^-7x) or (e^-7x)'.</p>
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<p>The<a>formula</a>we use to differentiate e^-7x is: d/dx (e^-7x) = -7e^-7x</p>
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<p>The<a>formula</a>we use to differentiate e^-7x is: d/dx (e^-7x) = -7e^-7x</p>
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<p>The formula applies for all x, as the exponential function is continuous everywhere.</p>
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<p>The formula applies for all x, as the exponential function is continuous everywhere.</p>
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<h2>Proofs of the Derivative of e^-7x</h2>
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<h2>Proofs of the Derivative of e^-7x</h2>
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<p>We can derive the derivative of e^-7x using proofs. To show this, we will use the rules of differentiation, particularly the chain rule. Here is how we can prove this:</p>
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<p>We can derive the derivative of e^-7x using proofs. To show this, we will use the rules of differentiation, particularly the chain rule. Here is how we can prove this:</p>
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<h3>Using the Chain Rule</h3>
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<h3>Using the Chain Rule</h3>
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<p>To prove the differentiation of e^-7x using the chain rule, We use the formula:</p>
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<p>To prove the differentiation of e^-7x using the chain rule, We use the formula:</p>
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<p>Let f(x) = e^u where u = -7x</p>
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<p>Let f(x) = e^u where u = -7x</p>
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<p>Using the chain rule: d/dx (e^u) = e^u · du/dx</p>
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<p>Using the chain rule: d/dx (e^u) = e^u · du/dx</p>
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<p>Therefore, d/dx (e^-7x) = e^-7x · d/dx(-7x) = e^-7x · (-7) = -7e^-7x</p>
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<p>Therefore, d/dx (e^-7x) = e^-7x · d/dx(-7x) = e^-7x · (-7) = -7e^-7x</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<h2>Higher-Order Derivatives of e^-7x</h2>
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<h2>Higher-Order Derivatives of e^-7x</h2>
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<p>When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.</p>
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<p>When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.</p>
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<p>To understand them better, consider how acceleration (second derivative) is the<a>rate</a>of change of velocity (first derivative). Higher-order derivatives can reveal the behavior of functions like e^-7x.</p>
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<p>To understand them better, consider how acceleration (second derivative) is the<a>rate</a>of change of velocity (first derivative). Higher-order derivatives can reveal the behavior of functions like e^-7x.</p>
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<p>For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.</p>
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<p>For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.</p>
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<p>For the nth Derivative of e^-7x, we generally use f^(n)(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).</p>
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<p>For the nth Derivative of e^-7x, we generally use f^(n)(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).</p>
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<h2>Special Cases:</h2>
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<h2>Special Cases:</h2>
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<p>Since e^-7x is an exponential function, it is continuous everywhere, and its derivative is defined for all x. The derivative is always negative, indicating that the function is decreasing for all<a>real numbers</a>.</p>
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<p>Since e^-7x is an exponential function, it is continuous everywhere, and its derivative is defined for all x. The derivative is always negative, indicating that the function is decreasing for all<a>real numbers</a>.</p>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of e^-7x</h2>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of e^-7x</h2>
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<p>Students frequently make mistakes when differentiating e^-7x. These mistakes can be resolved by understanding the correct methods. Here are a few common mistakes and ways to solve them:</p>
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<p>Students frequently make mistakes when differentiating e^-7x. These mistakes can be resolved by understanding the correct methods. Here are a few common mistakes and ways to solve them:</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>Calculate the derivative of (e^-7x · sin(x))</p>
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<p>Calculate the derivative of (e^-7x · sin(x))</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Here, we have f(x) = e^-7x · sin(x).</p>
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<p>Here, we have f(x) = e^-7x · sin(x).</p>
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<p>Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^-7x and v = sin(x).</p>
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<p>Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^-7x and v = sin(x).</p>
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<p>Let’s differentiate each term, u′ = d/dx (e^-7x) = -7e^-7x v′ = d/dx (sin(x)) = cos(x) Substituting into the given equation, f'(x) = (-7e^-7x) · sin(x) + e^-7x · cos(x)</p>
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<p>Let’s differentiate each term, u′ = d/dx (e^-7x) = -7e^-7x v′ = d/dx (sin(x)) = cos(x) Substituting into the given equation, f'(x) = (-7e^-7x) · sin(x) + e^-7x · cos(x)</p>
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<p>Let’s simplify terms to get the final answer, f'(x) = -7e^-7x · sin(x) + e^-7x · cos(x)</p>
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<p>Let’s simplify terms to get the final answer, f'(x) = -7e^-7x · sin(x) + e^-7x · cos(x)</p>
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<p>Thus, the derivative of the specified function is -7e^-7x · sin(x) + e^-7x · cos(x).</p>
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<p>Thus, the derivative of the specified function is -7e^-7x · sin(x) + e^-7x · cos(x).</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We find the derivative of the given function by dividing it into two parts. The first step is finding their derivatives and then combining them using the product rule to get the final result.</p>
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<p>We find the derivative of the given function by dividing it into two parts. The first step is finding their derivatives and then combining them using the product rule to get the final result.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>A scientist is measuring the decay of a radioactive substance, modeled by the function N(t) = e^-7t, where N(t) is the amount of substance remaining at time t. Find the rate of decay when t = 1 hour.</p>
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<p>A scientist is measuring the decay of a radioactive substance, modeled by the function N(t) = e^-7t, where N(t) is the amount of substance remaining at time t. Find the rate of decay when t = 1 hour.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We have N(t) = e^-7t (decay of the substance)...(1)</p>
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<p>We have N(t) = e^-7t (decay of the substance)...(1)</p>
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<p>Now, we will differentiate the equation (1) to find the rate of decay: dN/dt = -7e^-7t</p>
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<p>Now, we will differentiate the equation (1) to find the rate of decay: dN/dt = -7e^-7t</p>
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<p>Given t = 1 (substitute this into the derivative) dN/dt = -7e^-7(1) = -7e^-7</p>
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<p>Given t = 1 (substitute this into the derivative) dN/dt = -7e^-7(1) = -7e^-7</p>
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<p>Hence, the rate of decay at t = 1 hour is -7e^-7.</p>
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<p>Hence, the rate of decay at t = 1 hour is -7e^-7.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We find the rate of decay at t = 1 hour as -7e^-7, which represents the negative rate at which the substance decays over time.</p>
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<p>We find the rate of decay at t = 1 hour as -7e^-7, which represents the negative rate at which the substance decays over time.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>Derive the second derivative of the function y = e^-7x.</p>
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<p>Derive the second derivative of the function y = e^-7x.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The first step is to find the first derivative, dy/dx = -7e^-7x...(1)</p>
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<p>The first step is to find the first derivative, dy/dx = -7e^-7x...(1)</p>
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<p>Now we will differentiate equation (1) to get the second derivative:</p>
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<p>Now we will differentiate equation (1) to get the second derivative:</p>
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<p>d²y/dx² = d/dx [-7e^-7x] = -7 · d/dx [e^-7x] = -7(-7e^-7x) = 49e^-7x</p>
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<p>d²y/dx² = d/dx [-7e^-7x] = -7 · d/dx [e^-7x] = -7(-7e^-7x) = 49e^-7x</p>
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<p>Therefore, the second derivative of the function y = e^-7x is 49e^-7x.</p>
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<p>Therefore, the second derivative of the function y = e^-7x is 49e^-7x.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We use the step-by-step process, starting with the first derivative. By differentiating again, we find the second derivative using the constant multiple and chain rules.</p>
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<p>We use the step-by-step process, starting with the first derivative. By differentiating again, we find the second derivative using the constant multiple and chain rules.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Prove: d/dx (e^-7x²) = -14xe^-7x².</p>
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<p>Prove: d/dx (e^-7x²) = -14xe^-7x².</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Let’s start using the chain rule: Consider y = e^-7x²</p>
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<p>Let’s start using the chain rule: Consider y = e^-7x²</p>
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<p>To differentiate, we use the chain rule: dy/dx = e^-7x² · d/dx (-7x²) = e^-7x² · (-14x) = -14xe^-7x²</p>
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<p>To differentiate, we use the chain rule: dy/dx = e^-7x² · d/dx (-7x²) = e^-7x² · (-14x) = -14xe^-7x²</p>
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<p>Hence proved.</p>
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<p>Hence proved.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>In this step-by-step process, we used the chain rule to differentiate the equation. The derivative of the inner function -7x² is -14x, which we multiply with e^-7x² to get the result.</p>
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<p>In this step-by-step process, we used the chain rule to differentiate the equation. The derivative of the inner function -7x² is -14x, which we multiply with e^-7x² to get the result.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Solve: d/dx (e^-7x/x)</p>
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<p>Solve: d/dx (e^-7x/x)</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>To differentiate the function, we use the quotient rule: d/dx (e^-7x/x) = (d/dx (e^-7x) · x - e^-7x · d/dx(x))/x²</p>
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<p>To differentiate the function, we use the quotient rule: d/dx (e^-7x/x) = (d/dx (e^-7x) · x - e^-7x · d/dx(x))/x²</p>
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<p>We will substitute d/dx (e^-7x) = -7e^-7x and d/dx(x) = 1 = (-7e^-7x · x - e^-7x · 1)/x² = (-7xe^-7x - e^-7x)/x² = -7xe^-7x - e^-7x/x²</p>
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<p>We will substitute d/dx (e^-7x) = -7e^-7x and d/dx(x) = 1 = (-7e^-7x · x - e^-7x · 1)/x² = (-7xe^-7x - e^-7x)/x² = -7xe^-7x - e^-7x/x²</p>
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<p>Therefore, d/dx (e^-7x/x) = (-7xe^-7x - e^-7x)/x²</p>
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<p>Therefore, d/dx (e^-7x/x) = (-7xe^-7x - e^-7x)/x²</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.</p>
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<p>In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on the Derivative of e^-7x</h2>
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<h2>FAQs on the Derivative of e^-7x</h2>
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<h3>1.Find the derivative of e^-7x.</h3>
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<h3>1.Find the derivative of e^-7x.</h3>
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<p>Using the chain rule for e^-7x gives: d/dx (e^-7x) = -7e^-7x</p>
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<p>Using the chain rule for e^-7x gives: d/dx (e^-7x) = -7e^-7x</p>
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<h3>2.Can we use the derivative of e^-7x in real life?</h3>
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<h3>2.Can we use the derivative of e^-7x in real life?</h3>
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<p>Yes, we can use the derivative of e^-7x in real life to model<a>exponential decay</a>processes, such as radioactive decay or cooling of substances.</p>
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<p>Yes, we can use the derivative of e^-7x in real life to model<a>exponential decay</a>processes, such as radioactive decay or cooling of substances.</p>
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<h3>3.Is the derivative of e^-7x always negative?</h3>
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<h3>3.Is the derivative of e^-7x always negative?</h3>
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<p>Yes, the derivative of e^-7x is always negative, indicating a decreasing function for all x.</p>
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<p>Yes, the derivative of e^-7x is always negative, indicating a decreasing function for all x.</p>
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<h3>4.What rule is used to differentiate e^-7x/x?</h3>
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<h3>4.What rule is used to differentiate e^-7x/x?</h3>
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<p>We use the<a>quotient</a>rule to differentiate e^-7x/x, d/dx (e^-7x/x) = (x · (-7e^-7x) - e^-7x · 1)/x².</p>
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<p>We use the<a>quotient</a>rule to differentiate e^-7x/x, d/dx (e^-7x/x) = (x · (-7e^-7x) - e^-7x · 1)/x².</p>
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<h3>5.Are the derivatives of e^-7x and e^x the same?</h3>
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<h3>5.Are the derivatives of e^-7x and e^x the same?</h3>
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<p>No, they are different. The derivative of e^-7x is -7e^-7x, while the derivative of e^x is e^x.</p>
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<p>No, they are different. The derivative of e^-7x is -7e^-7x, while the derivative of e^x is e^x.</p>
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<h2>Important Glossaries for the Derivative of e^-7x</h2>
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<h2>Important Glossaries for the Derivative of e^-7x</h2>
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<ul><li><strong>Derivative:</strong>The derivative of a function indicates the rate at which the function changes in response to a change in x.</li>
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<ul><li><strong>Derivative:</strong>The derivative of a function indicates the rate at which the function changes in response to a change in x.</li>
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</ul><ul><li><strong>Exponential Function:</strong>Functions that have a constant base raised to a variable power, such as e^x.</li>
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</ul><ul><li><strong>Exponential Function:</strong>Functions that have a constant base raised to a variable power, such as e^x.</li>
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</ul><ul><li><strong>Chain Rule:</strong>A rule used to differentiate composite functions.</li>
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</ul><ul><li><strong>Chain Rule:</strong>A rule used to differentiate composite functions.</li>
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</ul><ul><li><strong>Constant Multiple Rule:</strong>Indicates how to differentiate functions that are multiplied by a constant.</li>
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</ul><ul><li><strong>Constant Multiple Rule:</strong>Indicates how to differentiate functions that are multiplied by a constant.</li>
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</ul><ul><li><strong>Quotient Rule:</strong>A rule used to differentiate functions that are divided by each other.</li>
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</ul><ul><li><strong>Quotient Rule:</strong>A rule used to differentiate functions that are divided by each other.</li>
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</ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
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</ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
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<h2>Jaskaran Singh Saluja</h2>
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<h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>