Question

How to differentiate

Answer 1

$\frac{dy}{dx}$ = 2a/b²(x + 1)

y = a/u² = a/ (b²/(x + 1)²) = a/b² (x + 1)²

differentiate using the ' chain rule'

$\frac{dy}{dx}$ = 2a/b² (x + 1)

Answer 2

First of all, substitute $u^2$ in the first expression:

$u = \dfrac{b}{x+1} \implies u^2 = \dfrac{b^2}{(x+1)^2}$

So, your function is

$y = \dfrac{a}{\frac{b^2}{(x+1)^2}} = \dfrac{a(x+1)^2}{b^2} = \dfrac{a}{b^2}(x+1)^2$

Since a and b are constants, we have

$\dfrac{dy}{dx} \dfrac{a}{b^2}(x+1)^2 = \dfrac{a}{b^2}\dfrac{dy}{dx}(x+1)^2$

The derivative of $(x+1)^2$ is

$2(x+1)\cdot 1$

following the rule

$\dfrac{d}{dx} f^n(x) = nf^{n-1}(x)f'(x)$

So, the answer is

$\dfrac{dy}{dx} \dfrac{a}{b^2}(x+1)^2 = \dfrac{2a}{b^2}(x+1)$