Question

if (f(x)=sin(x^{4})), what is (f(x)?)

Explanation:

Step1: Identify the outer - inner functions

Let $u = x^{4}$, then $y=\sin(u)$.

Step2: Differentiate the outer function

The derivative of $y = \sin(u)$ with respect to $u$ is $\frac{dy}{du}=\cos(u)$.

Step3: Differentiate the inner function

The derivative of $u = x^{4}$ with respect to $x$ is $\frac{du}{dx}=4x^{3}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\cos(u)$ and $\frac{du}{dx}=4x^{3}$ back in, and replace $u$ with $x^{4}$. So $\frac{dy}{dx}=\cos(x^{4})\cdot4x^{3}=4x^{3}\cos(x^{4})$.

Answer:

$4x^{3}\cos(x^{4})$