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

## Derivative of cos(e^{x})

**Question:** Find the derivative of cos(e^{x}).

*Answer: *

The derivative of cos(e^{x}) is equal to -e^{x}sin(e^{x}).

**Explanation:**

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

**Step 1:** We assume that z=e^{x}

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

**Step 3:** Now, by the chain rule, the derivative of cos(e^{x}) is equal to

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

= -sin z ⋅ e^{x} as we know that the derivative of sinx is cosx.

= -sin(e^{x}) ⋅ e^{x}, putting the value of z=e^{x}.

**Conclusion:** Hence, the derivative of cos(e^{x}) by the chain rule is -e^{x} sin(e^{x}).

**RELATED TOPICS:**

**Question:** What is the derivative of cos(e^{x}) at x=0?

*Answer:*

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

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

= [-e^{x} sin(e^{x})]_{{x=0}}

= -e^{0} sin(e^{0})

= – 1 ⋅ sin 1

= -sin 1

Thus, the derivative of cos(e^{x}) at the point x=0 is -sin 1.

## FAQs

**Q1: If y=cos(x ^{4}), then find dy/dx?**

**Answer:** If y=cos(e^{x}), then dy/dx= -e^{x} sin(e^{x}).