Derivative of e^sinx
2026-02-28 23:51 Diff
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Last updated on August 5, 2025

We use the derivative of e^sin(x), which combines the properties of the exponential and trigonometric functions, as a tool to measure how this complex function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of e^sin(x) in detail.

What is the Derivative of e^sinx?

We now understand the derivative of e^sinx. It is commonly represented as d/dx (e^sinx) or (e^sinx)', and its value is e^sinx * cosx.

The function e^sinx has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Exponential Function: (e^x), where e is the base of natural logarithms.

Chain Rule: Rule for differentiating e^sinx because it is a composition of functions.

Trigonometric Functions: sin(x) and cos(x) are used in the derivative.

Derivative of e^sinx Formula

The derivative of e^sinx can be denoted as d/dx (e^sinx) or (e^sinx)'.

The formula we use to differentiate e^sinx is: d/dx (e^sinx) = e^sinx * cosx

The formula applies to all x, as the exponential function and sine function are continuous everywhere.

Proofs of the Derivative of e^sinx

We can derive the derivative of e^sinx using proofs. To show this, we will use the chain rule along with the properties of exponential and trigonometric functions.

There are several methods we use to prove this, such as:

  1. Using Chain Rule
  2. Using Product Rule

We will now demonstrate that the differentiation of e^sinx results in e^sinx * cosx using the above-mentioned methods:

Using Chain Rule To prove the differentiation of e^sinx using the chain rule, We use the formula: Let u = sinx, then e^u = e^sinx

By the chain rule: d/dx [e^u] = e^u * du/dx u = sinx, so du/dx = cosx

Therefore, d/dx (e^sinx) = e^sinx * cosx Using Product Rule We will now prove the derivative of e^sinx using the product rule.

The step-by-step process is demonstrated below:

Here, we use the formula, Let y = e^sinx y = e^u, where u = sinx

The product rule formula: d/dx [u.v] = u'.v + u.v'

Since v = e^sinx is not a product, we use the chain rule as shown earlier.

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Higher-Order Derivatives of e^sinx

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.

o understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes.

Higher-order derivatives make it easier to understand functions like e^sinx. 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.

For the nth Derivative of e^sinx, we generally use fⁿ(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).

Special Cases:

When x is π/2, the derivative is e^sin(π/2) * 0 = 0 because cos(π/2) = 0. When x is 0, the derivative of e^sinx = e^0 * 1 = 1.

Common Mistakes and How to Avoid Them in Derivatives of e^sinx

Students frequently make mistakes when differentiating e^sinx. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:

Problem 1

Calculate the derivative of (e^sinx * ln(x))

Okay, lets begin

Here, we have f(x) = e^sinx * ln(x).

Using the product rule, f'(x) = u′v + uv′ In the given equation, u = e^sinx and v = ln(x).

Let’s differentiate each term, u′ = d/dx (e^sinx) = e^sinx * cosx v′ = d/dx (ln(x)) = 1/x Substituting into the given equation, f'(x) = (e^sinx * cosx) * ln(x) + (e^sinx) * (1/x)

Let’s simplify terms to get the final answer, f'(x) = e^sinx * cosx * ln(x) + e^sinx/x

Thus, the derivative of the specified function is e^sinx * cosx * ln(x) + e^sinx/x.

Explanation

We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.

Well explained 👍

Problem 2

XYZ Corporation measures a quantity represented by the function y = e^sin(x), where y represents a varying concentration at a distance x. If x = π/3 meters, calculate the rate of change of concentration.

Okay, lets begin

We have y = e^sin(x) (concentration function)...(1)

Now, we will differentiate the equation (1) Take the derivative of e^sin(x): dy/dx = e^sin(x) * cos(x)

Given x = π/3, substitute this into the derivative: dy/dx = e^sin(π/3) * cos(π/3) dy/dx = e^(√3/2) * 1/2

Hence, we get that the rate of change of concentration at x = π/3 is (e^(√3/2)) / 2.

Explanation

We find the rate of change of concentration at x = π/3 as (e^(√3/2)) / 2, indicating the rate at which concentration changes at this specific point.

Well explained 👍

Problem 3

Derive the second derivative of the function y = e^sin(x).

Okay, lets begin

The first step is to find the first derivative, dy/dx = e^sin(x) * cos(x)...(1)

Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [e^sin(x) * cos(x)]

Here we use the product rule, d²y/dx² = (d/dx [e^sin(x)]) * cos(x) + e^sin(x) * (d/dx [cos(x)])

d²y/dx² = (e^sin(x) * cos(x)) * cos(x) + e^sin(x) * (-sin(x))

d²y/dx² = e^sin(x) * cos²(x) - e^sin(x) * sin(x)

Therefore, the second derivative of the function y = e^sin(x) is e^sin(x) * (cos²(x) - sin(x)).

Explanation

We use the step-by-step process, where we start with the first derivative. Using the product rule, we differentiate e^sin(x) * cos(x). We then substitute the identity and simplify the terms to find the final answer.

Well explained 👍

Problem 4

Prove: d/dx (e^(2sin(x))) = 2e^(2sin(x)) * cos(x).

Okay, lets begin

Let’s start using the chain rule: Consider y = e^(2sin(x))

To differentiate, we use the chain rule: dy/dx = e^(2sin(x)) * d/dx (2sin(x)) = e^(2sin(x)) * 2cos(x)

Substituting y = e^(2sin(x)), d/dx (e^(2sin(x))) = 2e^(2sin(x)) * cos(x)

Hence proved.

Explanation

In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace 2sin(x) with its derivative. As a final step, we substitute y = e^(2sin(x)) to derive the equation.

Well explained 👍

Problem 5

Solve: d/dx (e^sinx/x)

Okay, lets begin

To differentiate the function, we use the quotient rule: d/dx (e^sinx/x) = (d/dx (e^sinx) * x - e^sinx * d/dx(x))/x²

We will substitute d/dx (e^sinx) = e^sinx * cosx and d/dx (x) = 1 (e^sinx * cosx * x - e^sinx * 1) / x² = (x * e^sinx * cosx - e^sinx) / x²

Therefore, d/dx (e^sinx/x) = (x * e^sinx * cosx - e^sinx) / x²

Explanation

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.

Well explained 👍

FAQs on the Derivative of e^sinx

1.Find the derivative of e^sinx.

Using the chain rule, the derivative of e^sinx gives e^sinx * cosx.

2.Can we use the derivative of e^sinx in real life?

Yes, we can use the derivative of e^sinx in real life in calculating rates of change in various scientific and engineering fields.

3.Is it possible to take the derivative of e^sinx at any point?

Yes, e^sinx is defined and differentiable at all real numbers, so it is possible to take the derivative at any point.

4.What rule is used to differentiate e^sinx/x?

We use the quotient rule to differentiate e^sinx/x, d/dx (e^sinx/x) = (x * e^sinx * cosx - e^sinx) / x².

5.Are the derivatives of e^sinx and ln(sinx) the same?

No, they are different. The derivative of e^sinx is e^sinx * cosx, while the derivative of ln(sinx) is cot(x).

6.Can we find the derivative of e^sinx using first principles?

While it is theoretically possible, it is more efficient to use the chain rule due to the complexity of e^sinx.

Important Glossaries for the Derivative of e^sinx

  • Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
  • Exponential Function: A mathematical function of the form e^x, where e is the base of natural logarithms.
  • Chain Rule: A rule in calculus for differentiating the compositions of functions.
  • Sine Function: A trigonometric function representing the y-coordinate of a point on the unit circle.
  • Product Rule: A rule for differentiating the product of two functions, stating d/dx [u.v] = u'.v + u.v'.

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Jaskaran Singh Saluja

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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.

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