Question

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

I have multipled the exponent with the number -2, giving -6.

-6(e^2x+1)^2

However not sure which rule to use in this question.

Thanks for the help!

Answer 1

Use the chain rule. Let

`y=-2(e^(2x)+1)^3`

and take
`u=e^(2x)+1`. The chain rule says

`(\mathrm dy)/(\mathrm dx)=(\mathrm dy)/(\mathrm du)\cdot(\mathrm du)/(\mathrm dx)`

The relevant derivatives are then

`(\mathrm dy)/(\mathrm du)=(\mathrm d(-2u^3))/(\mathrm du)=-6u^2`

(power rule)

`(\mathrm du)/(\mathrm dx)=(\mathrm d(e^(2x)+1))/(\mathrm dx)=2e^(2x)`

(chain rule applied to
`e^(2x)`; the constant vanishes)

So,

`(\mathrm dy)/(\mathrm dx)=-6u^2\cdot2e^(2x)=-12e^(2x)(e^(2x)+1)^2`