Question

let (f(x)=x^{3}) and (g(x)=\frac{x}{x - 1}). if (h) is the function defined by (h(x)=f(g(x))), which of the following gives a correct expression for (h(x))? a. (3(g(x))^{2}=3(\frac{x}{x - 1})^{2}) b. (3(g(x))^{2}=3(\frac{x}{x - 1})^{2}cdot\frac{- 1}{(x - 1)^{2}}) c. (3(g(x))^{2}g(x)=3(\frac{x}{x - 1})^{2}cdot\frac{-1}{(x - 1)^{2}}) d. ((g(x))^{3}=(\frac{x}{x - 1})^{3})

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$. Given $f(x)=x^{3}$, then $f^{\prime}(x) = 3x^{2}$.

Step2: Find $f^{\prime}(g(x))$

Substitute $x = g(x)$ into $f^{\prime}(x)$. Since $f^{\prime}(x)=3x^{2}$, then $f^{\prime}(g(x))=3(g(x))^{2}$.

Step3: Multiply by $g^{\prime}(x)$

We know that $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$. But we are not given $g^{\prime}(x)$ in the problem - statement for the general form of $h^{\prime}(x)$ in terms of the composition. If we assume we are just looking for the form of $f^{\prime}(g(x))$ (since the options seem to be focused on this part of the chain - rule application), the derivative of $h(x)=f(g(x))$ using the chain - rule gives $h^{\prime}(x)=3(g(x))^{2}\cdot g^{\prime}(x)$. If we consider the non - multiplied by $g^{\prime}(x)$ part (which might be a misinterpretation of the problem if $g^{\prime}(x)$ is non - 1, but looking at the options), the expression for the part related to $f^{\prime}(g(x))$ is $3(g(x))^{2}$.

Answer:

A. $3(g(x))^{2}$