The derivative of cos^{4}(x) is equal to -4cos^{3}(x) sinx. In this post, we will learn how to find the derivative of cos to the power 4 of x.

## Derivative of cos^{4}x Formula

The derivative of $\cos^4(x)$ is denoted by $\dfrac{d}{dx}(\cos^4 x)$ or $(\cos^4 x)’$. The formula of the derivative of cos^4(x) is given below:

$\dfrac{d}{dx}(\cos^4 x)=-4\cos^4 x \sin x$, or $(\cos^4 x)’=-4\cos^4 x \sin x$.

## Derivative of cos^{4}x

**Question:** What is the Derivative of $\cos^4x$?

*Answer:* The Derivative of $\cos^4x$ is $-4\cos^3 x \sin x$.

**Proof:**

**Step 1:** Let us assume that $z=\cos x$. Then we can write our function as

$\cos^4x=z^4$

**Step 2:** Note that $\dfrac{dz}{dx}=-\sin x$ as $z=\cos x$.

**Step 3:** By the chain rule, the derivative of $\cos^4x$ will be equal to

$\dfrac{d}{dx}(\cos^4x)=\dfrac{d}{dz}(z^4) \cdot \dfrac{dz}{dx}$

$=4z^3 \cdot (-\sin x)$ by the power rule of derivatives: $\dfrac{d}{dx}(x^n)=nx^{n-1}$

$=4\cos^3 x \cdot (-\sin x)$ as $z=\cos x$

$=-4\cos^3 x \sin x$.

**Conclusion:** The derivative of cos^4 x is -4cos^3(x) sinx and this is obtained by the chain rule and the power rule of derivatives.

In a similar way, one can obtain the derivative of \cos^n x, which is given below:

$\dfrac{d}{dx}(\cos^n x)=-n\cos^{n-1}x \sin x$.

For example,

- The derivative of $\cos^2x$ is $-2\cos x\sin x$.
- The derivative of $\cos^3x$ is $-2\cos^2 x\sin x$.
- The derivative of $\cos^5x$ is $-5\cos^4 x\sin x$.

**Also Read:**

## FAQs

**Q1: What is the derivative of cos^4x?**

**Answer:** The derivative of cos^{4}x is equal to -4cos^{3}x sinx.