Question

attempt 1: 10 attempts remaining. let $p(x)=5sqrt{x}+0.75x^{2}$. a. find the rate of change of $p(x), p(x)$. $p(x)=$ i b. find the rate of change of $p(x)$ per item produced, $\frac{p(x)}{x}$. $\frac{p(x)}{x}=$ i

Explanation:

Step1: Recall derivative rules

The derivative of $\sqrt{x}=x^{\frac{1}{2}}$ is $\frac{1}{2}x^{-\frac{1}{2}}$ by the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, and the derivative of $ax^n$ is $anx^{n - 1}$.

Step2: Differentiate $P(x)$

Given $P(x)=5\sqrt{x}+0.75x^{2}=5x^{\frac{1}{2}}+0.75x^{2}$. Then $P^{\prime}(x)=5\times\frac{1}{2}x^{-\frac{1}{2}}+0.75\times2x^{1}=\frac{5}{2\sqrt{x}} + 1.5x$.

Step3: Find $\frac{P^{\prime}(x)}{x}$

Substitute $P^{\prime}(x)$ into $\frac{P^{\prime}(x)}{x}$: $\frac{\frac{5}{2\sqrt{x}}+1.5x}{x}=\frac{5}{2x\sqrt{x}}+\frac{1.5x}{x}=\frac{5}{2x^{\frac{3}{2}}}+1.5$.

Answer:

a. $P^{\prime}(x)=\frac{5}{2\sqrt{x}} + 1.5x$
b. $\frac{P^{\prime}(x)}{x}=\frac{5}{2x^{\frac{3}{2}}}+1.5$