Differentiation using product rule

Question

I'm having trouble simplifying these questions, particularly when they involve square roots of $x$.

Differentiate the following with respect to $x$ and simplify:

$y=(x+2)x^\frac{3}{2}$

My attempt:

Using product rule: $u=x^\frac{3}{2}, v=(x+2)$ therefore $\frac{du}{dx}=\frac{3}{2}x^\frac{1}{2}, \frac{dv}{dx}=1\\$

$\frac{dy}{dx}= \frac{3\sqrt{x}}{2}(x+2)+(\sqrt{x})^3$

Factorise:$\sqrt{x}[\frac{3}{2}(x+2)+(\sqrt{x})^2]\\\sqrt{x}[\frac{3}{2}x+3+x]$

The given answer is $\frac{\sqrt{x}}{2}(5x+6)$, which I can't achieve and I can't understand why the denominator of 2 is a common factor.

Answer 1

You derivation is correct indeed note that

$$\sqrt{x}\left(\frac{3}{2}x+3+x\right)=\frac{\sqrt{x}}{2}\left(3x+6+2x\right)=\frac{\sqrt{x}}{2}\left(5x+6\right)$$

Answer 2

A general rule: first simplify, then differentiate. The function in question is a product, namely $$y=(x+2)x^\frac{3}{2}.$$ First simplify it to a sum $$y=x^{\frac{5}{2}}+2x^\frac{3}{2}$$ and then differentiate $$y’=\frac{5}{2}x^{\frac{3}{2}}+3x^{\frac12}.$$