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2026-01-01
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2026-02-28
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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 arctan(5x), which is 5/(1 + (5x)²), to understand how the arctangent function changes with respect to x. Derivatives are essential in calculating rates of change in various applications. We will explore the derivative of arctan(5x) in detail.</p>
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<p>We use the derivative of arctan(5x), which is 5/(1 + (5x)²), to understand how the arctangent function changes with respect to x. Derivatives are essential in calculating rates of change in various applications. We will explore the derivative of arctan(5x) in detail.</p>
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<h2>What is the Derivative of arctan(5x)?</h2>
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<h2>What is the Derivative of arctan(5x)?</h2>
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<p>To understand the derivative<a>of</a>arctan(5x), we represent it as d/dx (arctan(5x)) or (arctan(5x))'. The value of the derivative is 5/(1 + (5x)²). The<a>function</a>arctan(5x) is differentiable across its domain. The key concepts include:</p>
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<p>To understand the derivative<a>of</a>arctan(5x), we represent it as d/dx (arctan(5x)) or (arctan(5x))'. The value of the derivative is 5/(1 + (5x)²). The<a>function</a>arctan(5x) is differentiable across its domain. The key concepts include:</p>
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<p><strong>Arctangent Function:</strong>arctan(x) is the inverse of the tangent function. </p>
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<p><strong>Arctangent Function:</strong>arctan(x) is the inverse of the tangent function. </p>
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<p><strong>Chain Rule:</strong>Used to differentiate composite functions. </p>
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<p><strong>Chain Rule:</strong>Used to differentiate composite functions. </p>
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<p><strong>Standard Derivative:</strong>d/dx (arctan(u)) = u'/(1 + u²).</p>
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<p><strong>Standard Derivative:</strong>d/dx (arctan(u)) = u'/(1 + u²).</p>
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<h2>Derivative of arctan(5x) Formula</h2>
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<h2>Derivative of arctan(5x) Formula</h2>
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<p>The derivative of arctan(5x) is denoted as d/dx (arctan(5x)) or (arctan(5x))'.</p>
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<p>The derivative of arctan(5x) is denoted as d/dx (arctan(5x)) or (arctan(5x))'.</p>
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<p>The<a>formula</a>for differentiating arctan(5x) is: d/dx (arctan(5x)) = 5/(1 + (5x)²)</p>
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<p>The<a>formula</a>for differentiating arctan(5x) is: d/dx (arctan(5x)) = 5/(1 + (5x)²)</p>
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<p>This formula applies to all x.</p>
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<p>This formula applies to all x.</p>
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<h2>Proofs of the Derivative of arctan(5x)</h2>
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<h2>Proofs of the Derivative of arctan(5x)</h2>
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<p>We can prove the derivative of arctan(5x) using the chain rule. The proof involves differentiating the composite function using standard differentiation techniques. Here's how it's done:</p>
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<p>We can prove the derivative of arctan(5x) using the chain rule. The proof involves differentiating the composite function using standard differentiation techniques. Here's how it's done:</p>
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<h3>Using Chain Rule</h3>
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<h3>Using Chain Rule</h3>
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<p>To prove the differentiation of arctan(5x) using the chain rule, we start with: Let u = 5x, then arctan(5x) = arctan(u).</p>
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<p>To prove the differentiation of arctan(5x) using the chain rule, we start with: Let u = 5x, then arctan(5x) = arctan(u).</p>
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<p>The derivative of arctan(u) is d/dx (arctan(u)) = u'/(1 + u²).</p>
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<p>The derivative of arctan(u) is d/dx (arctan(u)) = u'/(1 + u²).</p>
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<p>Differentiating u = 5x gives u' = 5.</p>
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<p>Differentiating u = 5x gives u' = 5.</p>
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<p>Substitute into the derivative formula: d/dx (arctan(5x)) = 5/(1 + (5x)²).</p>
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<p>Substitute into the derivative formula: d/dx (arctan(5x)) = 5/(1 + (5x)²).</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 arctan(5x)</h2>
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<h2>Higher-Order Derivatives of arctan(5x)</h2>
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<p>Higher-order derivatives of functions like arctan(5x) can provide insights into the function's behavior. Differentiating arctan(5x)<a>multiple</a>times yields higher-order derivatives.</p>
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<p>Higher-order derivatives of functions like arctan(5x) can provide insights into the function's behavior. Differentiating arctan(5x)<a>multiple</a>times yields higher-order derivatives.</p>
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<p>For instance, the first derivative gives the<a>rate</a>of change, while the second derivative indicates how this rate changes.</p>
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<p>For instance, the first derivative gives the<a>rate</a>of change, while the second derivative indicates how this rate changes.</p>
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<p>For the first derivative, we write y′(x). The second derivative, y′′(x), is derived from the first derivative, and this pattern continues.</p>
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<p>For the first derivative, we write y′(x). The second derivative, y′′(x), is derived from the first derivative, and this pattern continues.</p>
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<p>For the nth derivative, we use yⁿ(x).</p>
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<p>For the nth derivative, we use yⁿ(x).</p>
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<h2>Special Cases:</h2>
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<h2>Special Cases:</h2>
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<p>When x is 0, the derivative of arctan(5x) is 5/(1 + (5×0)²) = 5.</p>
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<p>When x is 0, the derivative of arctan(5x) is 5/(1 + (5×0)²) = 5.</p>
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<p>As x approaches infinity, the derivative approaches 0 because the<a>denominator</a>grows much larger than the<a>numerator</a>.</p>
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<p>As x approaches infinity, the derivative approaches 0 because the<a>denominator</a>grows much larger than the<a>numerator</a>.</p>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of arctan(5x)</h2>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of arctan(5x)</h2>
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<p>Students often make mistakes when differentiating arctan(5x). Understanding the correct process can help avoid these errors. Here are some common mistakes and solutions:</p>
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<p>Students often make mistakes when differentiating arctan(5x). Understanding the correct process can help avoid these errors. Here are some common mistakes and solutions:</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 arctan(5x)·x².</p>
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<p>Calculate the derivative of arctan(5x)·x².</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 f(x) = arctan(5x)·x². Using the product rule, f'(x) = u′v + uv′. Let u = arctan(5x) and v = x².</p>
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<p>We have f(x) = arctan(5x)·x². Using the product rule, f'(x) = u′v + uv′. Let u = arctan(5x) and v = x².</p>
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<p>Differentiate each term: u′ = 5/(1 + (5x)²) v′ = 2x</p>
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<p>Differentiate each term: u′ = 5/(1 + (5x)²) v′ = 2x</p>
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<p>Substitute into the product rule: f'(x) = (5/(1 + (5x)²))·x² + arctan(5x)·2x</p>
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<p>Substitute into the product rule: f'(x) = (5/(1 + (5x)²))·x² + arctan(5x)·2x</p>
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<p>Thus, the derivative of the function is (5x²)/(1 + (5x)²) + 2x arctan(5x).</p>
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<p>Thus, the derivative of the function is (5x²)/(1 + (5x)²) + 2x arctan(5x).</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We differentiate the function by dividing it into two parts and applying the product rule. This involves differentiating each part separately and combining the results.</p>
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<p>We differentiate the function by dividing it into two parts and applying the product rule. This involves differentiating each part separately and combining the results.</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 car's acceleration is modeled by a = arctan(5x). Find the rate of change of acceleration when x = 1.</p>
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<p>A car's acceleration is modeled by a = arctan(5x). Find the rate of change of acceleration when x = 1.</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 a = arctan(5x) (acceleration model)...(1)</p>
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<p>We have a = arctan(5x) (acceleration model)...(1)</p>
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<p>Differentiate equation (1) to find the rate of change of acceleration: da/dx = 5/(1 + (5x)²)</p>
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<p>Differentiate equation (1) to find the rate of change of acceleration: da/dx = 5/(1 + (5x)²)</p>
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<p>Substitute x = 1 into the derivative: da/dx = 5/(1 + (5×1)²) = 5/26</p>
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<p>Substitute x = 1 into the derivative: da/dx = 5/(1 + (5×1)²) = 5/26</p>
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<p>Hence, the rate of change of acceleration when x = 1 is 5/26.</p>
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<p>Hence, the rate of change of acceleration when x = 1 is 5/26.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>By differentiating the acceleration function, we find the rate at which acceleration changes at a specific point, x = 1, giving us a value of 5/26.</p>
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<p>By differentiating the acceleration function, we find the rate at which acceleration changes at a specific point, x = 1, giving us a value of 5/26.</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 = arctan(5x).</p>
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<p>Derive the second derivative of the function y = arctan(5x).</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Find the first derivative: dy/dx = 5/(1 + (5x)²)...(1)</p>
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<p>Find the first derivative: dy/dx = 5/(1 + (5x)²)...(1)</p>
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<p>Differentiate equation (1) for the second derivative: d²y/dx² = d/dx [5/(1 + (5x)²)]</p>
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<p>Differentiate equation (1) for the second derivative: d²y/dx² = d/dx [5/(1 + (5x)²)]</p>
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<p>Apply the quotient rule: d²y/dx² = -50x/(1 + (5x)²)²</p>
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<p>Apply the quotient rule: d²y/dx² = -50x/(1 + (5x)²)²</p>
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<p>Therefore, the second derivative of y = arctan(5x) is -50x/(1 + (5x)²)².</p>
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<p>Therefore, the second derivative of y = arctan(5x) is -50x/(1 + (5x)²)².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the quotient rule, we differentiate the first derivative to obtain the second derivative, simplifying the result to find the final answer.</p>
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<p>Using the quotient rule, we differentiate the first derivative to obtain the second derivative, simplifying the result to find the final answer.</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 (arctan²(5x)) = 10x arctan(5x)/(1 + (5x)²).</p>
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<p>Prove: d/dx (arctan²(5x)) = 10x arctan(5x)/(1 + (5x)²).</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Use the chain rule: Let y = arctan²(5x) = [arctan(5x)]²</p>
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<p>Use the chain rule: Let y = arctan²(5x) = [arctan(5x)]²</p>
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<p>Differentiate using the chain rule: dy/dx = 2[arctan(5x)]·d/dx[arctan(5x)] With d/dx[arctan(5x)] = 5/(1 + (5x)²), dy/dx = 2[arctan(5x)]·5/(1 + (5x)²)</p>
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<p>Differentiate using the chain rule: dy/dx = 2[arctan(5x)]·d/dx[arctan(5x)] With d/dx[arctan(5x)] = 5/(1 + (5x)²), dy/dx = 2[arctan(5x)]·5/(1 + (5x)²)</p>
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<p>Simplify: dy/dx = 10x arctan(5x)/(1 + (5x)²) Hence, proved.</p>
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<p>Simplify: dy/dx = 10x arctan(5x)/(1 + (5x)²) Hence, proved.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the chain rule, we differentiate the square of arctan(5x) and substitute the derivative of arctan(5x) to prove the equation.</p>
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<p>Using the chain rule, we differentiate the square of arctan(5x) and substitute the derivative of arctan(5x) to prove the equation.</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 (arctan(5x)/x).</p>
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<p>Solve: d/dx (arctan(5x)/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>Differentiate using the quotient rule: d/dx (arctan(5x)/x) = (d/dx (arctan(5x))·x - arctan(5x)·d/dx(x))/x²</p>
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<p>Differentiate using the quotient rule: d/dx (arctan(5x)/x) = (d/dx (arctan(5x))·x - arctan(5x)·d/dx(x))/x²</p>
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<p>Substitute d/dx (arctan(5x)) = 5/(1 + (5x)²) and d/dx(x) = 1: = (5x/(1 + (5x)²) - arctan(5x))/x² = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²))</p>
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<p>Substitute d/dx (arctan(5x)) = 5/(1 + (5x)²) and d/dx(x) = 1: = (5x/(1 + (5x)²) - arctan(5x))/x² = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²))</p>
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<p>Therefore, d/dx (arctan(5x)/x) = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²)).</p>
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<p>Therefore, d/dx (arctan(5x)/x) = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²)).</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Using the quotient rule, we differentiate arctan(5x)/x, substituting the derivatives and simplifying to obtain the final result.</p>
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<p>Using the quotient rule, we differentiate arctan(5x)/x, substituting the derivatives and simplifying 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 arctan(5x)</h2>
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<h2>FAQs on the Derivative of arctan(5x)</h2>
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<h3>1.Find the derivative of arctan(5x).</h3>
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<h3>1.Find the derivative of arctan(5x).</h3>
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<p>Using the chain rule, the derivative of arctan(5x) is 5/(1 + (5x)²).</p>
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<p>Using the chain rule, the derivative of arctan(5x) is 5/(1 + (5x)²).</p>
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<h3>2.Can we use the derivative of arctan(5x) in real life?</h3>
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<h3>2.Can we use the derivative of arctan(5x) in real life?</h3>
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<p>Yes, the derivative of arctan(5x) can model rates of change in physics, engineering, and other fields where arctangent relationships occur.</p>
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<p>Yes, the derivative of arctan(5x) can model rates of change in physics, engineering, and other fields where arctangent relationships occur.</p>
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<h3>3.Is it possible to take the derivative of arctan(5x) at any point?</h3>
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<h3>3.Is it possible to take the derivative of arctan(5x) at any point?</h3>
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<p>Yes, the derivative of arctan(5x) is valid for all real x.</p>
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<p>Yes, the derivative of arctan(5x) is valid for all real x.</p>
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<h3>4.What rule is used to differentiate arctan(5x)/x?</h3>
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<h3>4.What rule is used to differentiate arctan(5x)/x?</h3>
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<p>The<a>quotient</a>rule is used to differentiate arctan(5x)/x: d/dx (arctan(5x)/x) = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²)).</p>
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<p>The<a>quotient</a>rule is used to differentiate arctan(5x)/x: d/dx (arctan(5x)/x) = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²)).</p>
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<h3>5.How does the derivative of arctan(5x) compare to arctan(x)?</h3>
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<h3>5.How does the derivative of arctan(5x) compare to arctan(x)?</h3>
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<p>The derivative of arctan(5x) is 5/(1 + (5x)²), while the derivative of arctan(x) is 1/(1 + x²).</p>
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<p>The derivative of arctan(5x) is 5/(1 + (5x)²), while the derivative of arctan(x) is 1/(1 + x²).</p>
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<h3>6.Can we find the derivative of the arctan(5x) formula?</h3>
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<h3>6.Can we find the derivative of the arctan(5x) formula?</h3>
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<p>Yes, using the chain rule: d/dx (arctan(5x)) = 5/(1 + (5x)²).</p>
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<p>Yes, using the chain rule: d/dx (arctan(5x)) = 5/(1 + (5x)²).</p>
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<h2>Important Glossaries for the Derivative of arctan(5x)</h2>
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<h2>Important Glossaries for the Derivative of arctan(5x)</h2>
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<ul><li><strong>Derivative:</strong>Indicates how a function changes with respect to a variable.</li>
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<ul><li><strong>Derivative:</strong>Indicates how a function changes with respect to a variable.</li>
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</ul><ul><li><strong>Arctangent Function:</strong>The inverse of the tangent function, denoted as arctan(x).</li>
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</ul><ul><li><strong>Arctangent Function:</strong>The inverse of the tangent function, denoted as arctan(x).</li>
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</ul><ul><li><strong>Chain Rule:</strong>A rule for differentiating composite functions.</li>
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</ul><ul><li><strong>Chain Rule:</strong>A rule for differentiating composite functions.</li>
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</ul><ul><li><strong>Quotient Rule:</strong>A rule used to differentiate ratios of functions.</li>
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</ul><ul><li><strong>Quotient Rule:</strong>A rule used to differentiate ratios of functions.</li>
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</ul><ul><li><strong>Second Derivative:</strong>The derivative of the derivative, showing how the rate of change itself changes.</li>
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</ul><ul><li><strong>Second Derivative:</strong>The derivative of the derivative, showing how the rate of change itself changes.</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>