section 3.4 the chain question 16, 3.4.51 find the derivative. f(x) if f(x)=(e^{x^{4}+1})^{3} f(x)=□

Explanation:

Step1: Apply chain - rule

Let $u = e^{x^{4}}+1$, then $F(x)=u^{3}$. By the chain - rule $\frac{dF}{dx}=\frac{dF}{du}\cdot\frac{du}{dx}$. First, find $\frac{dF}{du}$. Since $F = u^{3}$, then $\frac{dF}{du}=3u^{2}=3(e^{x^{4}} + 1)^{2}$.

Step2: Find $\frac{du}{dx}$

Let $v=x^{4}$, then $u = e^{v}+1$. By the chain - rule $\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}$. Since $\frac{du}{dv}=e^{v}$ and $\frac{dv}{dx}=4x^{3}$, then $\frac{du}{dx}=e^{x^{4}}\cdot4x^{3}$.

Step3: Calculate $\frac{dF}{dx}$

$\frac{dF}{dx}=\frac{dF}{du}\cdot\frac{du}{dx}=3(e^{x^{4}} + 1)^{2}\cdot e^{x^{4}}\cdot4x^{3}=12x^{3}e^{x^{4}}(e^{x^{4}} + 1)^{2}$.

Answer:

$12x^{3}e^{x^{4}}(e^{x^{4}} + 1)^{2}$