Answer:
\(e^p(p+p\sqrt{p}+1+\frac{3}{2}\sqrt{p})\)
Step-by-step explanation:
We are given the following information in the question:
\(y = e^p(p+p\sqrt{p})\\\\y = e^p(p + p^{\frac{3}{2}})\)
We will use the product rule to differentiate the above expression.
The product rule says that:
\(\displaystyle\frac{d(uv)}{dx} = u.\frac{dv}{dx} + v \frac{du}{dx}\)
The differentiation is done in the following ways:
\(\displaystyle\frac{dy}{dx} = \frac{d(e^p)}{dp}(p+p\sqrt{p}) + e^p\frac{d(p+p\sqrt{p})}{dp}\\\\= e^p(p+p\sqrt{p}) + e^p(1+\frac{3}{2}\sqrt{p})\\\\= e^p(p+p\sqrt{p}+1+\frac{3}{2}\sqrt{p})\)
The above is the required differentiation of the given expression.
