Question

question 20. 1 point determine $p(x)$ when $p(x)=\frac{0.08e^{x}}{sqrt{x}}$. select the correct answer below: $p(x)=0.08\frac{1}{sqrt2{x}}(e^{x})(sqrt{x}) - \frac{1}{sqrt2{x}}(e^{x})(sqrt{x})$, $p(x)=0.08\frac{(xe^{x - 1})(sqrt{x}) - (e^{x})(sqrt{x})}{sqrt2{x}}$, $p(x)=0.08\frac{(e^{x})(\frac{1}{sqrt2{x}})-(sqrt{x})(e^{x})}{sqrt2{x}}$, $p(x)=\frac{0.08e^{x}}{sqrt2{x}} - \frac{1}{sqrt2{x}}$

Explanation:

Step1: Apply quotient - rule for differentiation.

If $p(x)=\frac{u(x)}{v(x)}$, then $p'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$. Here, $u(x) = 0.08e^{x}$ and $v(x)=\sqrt{x}$.

Step2: Find $u'(x)$ and $v'(x)$.

$u'(x)=0.08e^{x}$, $v'(x)=\frac{1}{2\sqrt{x}}$.

Step3: Substitute into quotient - rule.

$p'(x)=\frac{0.08e^{x}\cdot\sqrt{x}-0.08e^{x}\cdot\frac{1}{2\sqrt{x}}}{x}=\frac{0.08e^{x}(\sqrt{x}-\frac{1}{2\sqrt{x}})}{x}=\frac{0.08e^{x}(\frac{2x - 1}{2\sqrt{x}})}{x}=\frac{0.08e^{x}(2x - 1)}{2x\sqrt{x}}$.

Answer:

The correct derivative formula based on the above steps. (It seems the options are not clearly readable in the provided image to pick a specific one from the list).