# Derivative of cos(e^x) by Chain Rule

The derivative of cos(ex) is equal to -ex sin(ex). In this post, we will learn how to find the derivative of cos(ex) by the chain rule of derivatives.

## Derivative of cos(ex)

Question: Find the derivative of cos(ex).

The derivative of cos(ex) is equal to -exsin(ex).

Explanation:

Note that f(x)=cos(ex) is a composite function. The following steps are needed to find the derivative of cos(ex) using the chain rule of derivatives.

Step 1: We assume that z=ex

Step 2: We have that  $\dfrac{dz}{dx}=e^x$.

Step 3: Now, by the chain rule, the derivative of cos(ex) is equal to

$\dfrac{d}{dx}(\cos e^x)=\dfrac{d}{dz}(\cos z) \cdot \dfrac{dz}{dx}$

= -sin z ⋅ ex as we know that the derivative of sinx is cosx.

= -sin(ex) ⋅ ex, putting the value of z=ex.

Conclusion: Hence, the derivative of cos(ex) by the chain rule is -ex sin(ex).

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Derivative of cos(x4)

Question: What is the derivative of cos(ex) at x=0?

From the above, we have obtained that the derivative of cos(ex) is -ex sin(ex). Thus, at the point x=0 we have that

$\dfrac{d}{dx}[\cos(e^x)]$

= [-ex sin(ex)]{x=0}

= -e0 sin(e0)

= – 1 ⋅ sin 1

= -sin 1

Thus, the derivative of cos(ex) at the point x=0 is -sin 1.

## FAQs

Q1: If y=cos(x4), then find dy/dx?

Answer: If y=cos(ex), then dy/dx= -ex sin(ex).