Question

Find the derivative of (1+2x)^x^2

Answer 1

`(1+2x)^(x^2)=e^{\ln(1+2x)^(x^2)}=e^(x^2\ln(1+2x))`

Differentiate with the chain rule:


`(\mathrm d)/(\mathrm dx)e^(x^2\ln(1+2x))=e^(x^2\ln(1+2x))(\mathrm d)/(\mathrm dx)[x^2\ln(1+2x)]`

Use the product rule for the rest:


`e^(x^2\ln(1+2x))\left(2x\ln(1+2x)+(2x^2)/(1+2x)\right)=2x\left(\ln(1+2x)+\frac x{1+2x}\right)(1+2x)^(x^2)`