algebra calculus question

Question

If $$y= \frac{\sqrt{x}}{2x+3 }$$ what will be the answer to this question if we have to differentiate it with respect to $x$?

Answer 1

you have: $$y= \frac{√x}{2x+3 }$$ Use division rule:

for $y=\frac{u}{v}$, for derivative with respect to x:$$\frac{dy}{dx}=\frac{v\cdot \frac{du}{dx}-u\cdot \frac{dv}{dx}}{v^2}$$

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

$$=\frac{(2x+3)\cdot \frac 1 {2\sqrt x}-\sqrt x\cdot 2 }{(2x+3)^2}$$

$$=\frac{2\sqrt x \cdot (2x+3)-4x}{2\sqrt x\cdot (2x+3)^2}$$