Question

1. let ( f(x)=5 x^{4} ) and ( g(x)=e^{2 x}+x ). if ( h ) is the function defined by ( h(x)=f(g(x)) ), which of the following gives a correct expression for ( h^{prime}(x) )?

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(h(x)=f(g(x))\), then \(h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)\).
First, find \(f^{\prime}(x)\) and \(g^{\prime}(x)\).
For \(f(x) = 5x^{4}\), using the power rule \((x^{n})^{\prime}=nx^{n - 1}\), we have \(f^{\prime}(x)=20x^{3}\).
For \(g(x)=e^{2x}+x\), using the sum rule \((u + v)^{\prime}=u^{\prime}+v^{\prime}\) and the chain rule for \(u = e^{2x}\) (\((e^{u})^{\prime}=e^{u}\cdot u^{\prime}\), here \(u = 2x\), \(u^{\prime}=2\)) and \((x)^{\prime}=1\), we get \(g^{\prime}(x)=2e^{2x}+1\).

Step2: Substitute into the chain - rule formula

Substitute \(g(x)\) into \(f^{\prime}(x)\) and multiply by \(g^{\prime}(x)\).
\(f^{\prime}(g(x)) = 20(e^{2x}+x)^{3}\), and \(g^{\prime}(x)=2e^{2x}+1\).
So \(h^{\prime}(x)=20(e^{2x}+x)^{3}(2e^{2x}+1)\)

Answer:

\(h^{\prime}(x)=20(e^{2x}+x)^{3}(2e^{2x}+1)\)