find the derivative of ( f(x) ).\n\n( f(x)=e^{x^{2}} )\n\n( f^{prime}(x)= )
find the derivative of ( f(x) ).\n\n( f(x)=e^{x^{2}} )\n\n( f^{prime}(x)= )
Answer
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}})