Question

find the derivative of ( f(x) ).

( f(x)=e^{x^{2}} )

( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(g(x))\), then \(y^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Let \(u = x^{2}\), so \(f(x)=e^{u}\). The derivative of \(e^{u}\) with respect to \(u\) is \(e^{u}\), and the derivative of \(u = x^{2}\) with respect to \(x\) is \(2x\).

Step2: Substitute back

By the chain rule, \(f^\prime(x)=\frac{d}{du}(e^{u})\cdot\frac{du}{dx}\). Substituting \(u = x^{2}\) back in, we get \(f^\prime(x)=e^{x^{2}}\cdot2x\).

Answer:

\(2x e^{x^{2}}\)