Question

How to find first Derivative of -2(e^2x+1)^3?

Answer 1

Use the chain rule. Let

$y=-2(e^{2x}+1)^3$

and take $u=e^{2x}+1$. The chain rule says

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm du}\cdot\dfrac{\mathrm du}{\mathrm dx}$

The relevant derivatives are then

$\dfrac{\mathrm dy}{\mathrm du}=\dfrac{\mathrm d(-2u^3)}{\mathrm du}=-6u^2$

(power rule)

$\dfrac{\mathrm du}{\mathrm dx}=\dfrac{\mathrm d(e^{2x}+1)}{\mathrm dx}=2e^{2x}$

(chain rule applied to $e^{2x}$; the constant vanishes)

So,

$\dfrac{\mathrm dy}{\mathrm dx}=-6u^2\cdot2e^{2x}=-12e^{2x}(e^{2x}+1)^2$