Question

find the derivative of ( f(x) ).
( f(x)=e^{x^{4}} )
( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = e^{u}\) and \(u = x^{4}\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
For \(y = e^{u}\), \(\frac{dy}{du}=e^{u}\).
For \(u = x^{4}\), \(\frac{du}{dx}=4x^{3}\).

Step2: Substitute back

Substitute \(u = x^{4}\) into \(\frac{dy}{du}\cdot\frac{du}{dx}\).
We get \(e^{x^{4}}\cdot4x^{3}\).

Answer:

\(4x^{3}e^{x^{4}}\)