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