How to simplify this derivative?

Question

I don't understand how I can go from the first step to the step on the right, can someone help me please? $$ \frac{dy}{dx} = \frac{3 \sqrt{x}}{2\sqrt{1+x^2}} - \frac{\sqrt{x^5}}{\sqrt{(1+x^2)^3}} = \frac{\sqrt{x}(3+x^2)}{2\sqrt{(1+x^2)^3}} $$

Answer 1

Hint: $$ \sqrt{1+x^2} \, (1+x^2) = \sqrt{(1+x^2)^3} $$

Use $$ \frac{a}{b} - \frac{c}{d} = \frac{da - cb}{bd} $$

and $$ \frac{a b}{a c} = \frac{b}{c} $$

Answer 2

Hint: For $x \ge 0$, we have $\sqrt{x^5}=x^2\sqrt x$, and $\sqrt{(1+x^2)^3} = (1+x^2)\sqrt{1+x^2}$.

Answer 3

Suppose you begin by taking out a factor $\cfrac {\sqrt x}{\sqrt{(1+x^2)^3}}$ to obtain: $$\cfrac {\sqrt x}{\sqrt{(1+x^2)^3}}\cdot \left(\frac {3(1+x^2)}{2}-x^2\right)$$

That depends on how confident you are with the square-root expressions involved. It is a useful technique to note with computations like this which seem a bit complicated - see if you can extract some of the complexity to leave a simpler computation.