Question

Question #042b9

Answer 1

`f'(x)=(2x)^x(ln(2x)+1)`

Explanation:

`f(x)=(2x)^x`

Take the natural logarithm of both sides:

`ln(f(x))=ln((2x)^x)`

The right-hand side can be rewritten using `ln(a^b)=bln(a)`:

`ln(f(x))=xln(2x)`

Differentiate both sides of the equation. The chain and product rules will be used.

`1/f(x)f'(x)=ln(2x)(d/dxx)+x(d/dxln(2x))`

`1/(2x)^xf'(x)=ln(2x)+x(1/(2x))(d/dx2x)`

`1/(2x)^xf'(x)=ln(2x)+1`

`f'(x)=(2x)^x(ln(2x)+1)`