Question

the function h(x) is defined in terms of a differentiable f(x). when h(x) = \frac{f(x^{5})}{x}, find an expression for h(x). choose the correct answer below. a. h(x)=\frac{4x^{5}f(x^{4}) - f(x^{5})}{x^{2}} b. h(x)=\frac{4x^{5}f(x^{5}) - f(x^{4})}{x^{2}} c. h(x)=\frac{5x^{5}f(x^{5}) - f(x^{5})}{x^{2}} d. h(x)=\frac{5x^{4}f(x^{4}) - f(x^{5})}{x^{2}}

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $h(x)=\frac{u(x)}{v(x)}$, then $h^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v^{2}(x)}$. Here, $u(x)=f(x^{5})$ and $v(x)=x$.

Step2: Find $u^{\prime}(x)$ using chain - rule

If $u(x)=f(x^{5})$, by the chain - rule, if $y = f(u)$ and $u=x^{5}$, then $\frac{dy}{dx}=f^{\prime}(u)\cdot\frac{du}{dx}$. Since $\frac{du}{dx}=5x^{4}$, $u^{\prime}(x)=f^{\prime}(x^{5})\cdot5x^{4}$. And $v^{\prime}(x) = 1$.

Step3: Substitute into quotient - rule formula

$h^{\prime}(x)=\frac{f^{\prime}(x^{5})\cdot5x^{4}\cdot x - f(x^{5})\cdot1}{x^{2}}=\frac{5x^{5}f^{\prime}(x^{5})-f(x^{5})}{x^{2}}$

Answer:

C. $h^{\prime}(x)=\frac{5x^{5}f^{\prime}(x^{5})-f(x^{5})}{x^{2}}$