Question

Differentiate e^(sin^25x)

Answer 1

To differentiate \( e^{\sin^2(5x)} \), we will use the chain rule.

Let \( y = e^{\sin^2(5x)} \). The derivative can be found as follows:

1. Differentiate the outer function \( e^u \) where \( u = \sin^2(5x) \):
\[
\frac{dy}{du} = e^u = e^{\sin^2(5x)}
\]

2. Next, differentiate the inner function \( u = \sin^2(5x) \):
- First, apply the chain rule again for \( \sin^2(5x) \):
\[
\frac{du}{dx} = 2\sin(5x) \cdot \frac{d}{dx}(\sin(5x))
\]
- Now differentiate \( \sin(5x) \):
\[
\frac{d}{dx}(\sin(5x)) = 5\cos(5x)
\]
- Putting it all together:
\[
\frac{du}{dx} = 2\sin(5x) \cdot 5\cos(5x) = 10\sin(5x)\cos(5x)
\]

3. Now, apply the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^{\sin^2(5x)} \cdot 10\sin(5x)\cos(5x)
\]

4. Finally, we can rewrite \( 10\sin(5x)\cos(5x) \) using the double angle identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \):
\[
10\sin(5x)\cos(5x) = 5\sin(10x)
\]

Putting it all together, the derivative is:
\[
\frac{dy}{dx} = 5e^{\sin^2(5x)} \sin(10x)
\]

Thus, the final result for the derivative of \( e^{\sin^2(5x)} \) is:
\[
\frac{d}{dx}(e^{\sin^2(5x)}) = 5e^{\sin^2(5x)} \sin(10x)
\]