[Solved] What is the Derivative of cos(x^4)?

To find the derivative of cos(x^4), we will use the chain rule of derivatives. The chain rule is used to find the derivative of a composite function which we recall now: The derivative of a composite function f(g(x)) by chain rule is given by

 

$\dfrac{d}{dx}(f(g(x)))$ $=f'(g(x)) \cdot g'(x)$ where the prime $’$ denotes the first-order derivative.

 

Let $f(x)=\cos x$ and $g(x)=x^4$.

 

We have: $f'(x)=-\sin x$ and $g'(x)=4x^3$.

 

This implies that $f'(g(x))=f'(x^4)=-\sin(x^4)$ as the derivative of $\cos x$ is $-\sin x$.

 

Now, by the above chain rule, the derivative of $\cos x^4$ is equal to

 

$\dfrac{d}{dx}(\cos x^4)$ $=f'(g(x)) \cdot g'(x)$

 

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

 

$=-4x^3\sin(x^4)$.

 

Thus, the derivative of cos(x^4) is equal to -4x^3 sin(x^4). This is proved by the chain rule of derivatives.

 

Question-Answer on Derivative of cos(x^4)

Question 1: Find the derivative of cos(x^4) at x=0.

 

 

From the above, the derivative of cos(x^4) is equal to -4x^3 sin(x^4). So, at $x=0$ we have

 

$\dfrac{d}{dx}[\cos(x^4)]$

 

$=[-4x^3 \sin(x^4)]_{x=0}$

 

$=[-4 \cdot 0^3 \sin(0)]$

 

$=0$

 

Hence, the derivative of cos(x^4) at the point x=0 is equal to zero.