Question

let ( f(x)=5x^{4}) and ( g(x)=e^{2x}+x). if ( h(x)=f(g(x))), which of the following gives a correct expression for ( h(x))? a) (20(g(x))^{3}(20(e^{2x}+x)^{3})) b) (20(g(x))^{3}(20(2e^{2x}+1)^{3})) c) (20(g(x))^{3}cdot g(x)(20(e^{2x}+x)^{3}cdot(2e^{2x}+1))) d) (5(g(x))^{4}(5(2e^{2x}+1)^{4}))

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$.

Step2: Differentiate $f(x)$

Given $f(x)=5x^{4}$, then $f^{\prime}(x)=20x^{3}$ by the power rule $\frac{d}{dx}(x^{n}) = nx^{n - 1}$.

Step3: Substitute $g(x)$ into $f^{\prime}(x)$ and multiply by $g^{\prime}(x)$

We have $g(x)=e^{2x}+x$, so $g^{\prime}(x)=2e^{2x}+1$. Then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)=20(g(x))^{3}\cdot(2e^{2x}+1)=20(e^{2x}+x)^{3}\cdot(2e^{2x}+1)$.

Answer:

C. $20(e^{2x}+x)^{3}\cdot(2e^{2x}+1)$