Let f ( x ) = x ^(2 x) . Use logarithmic differentiation to determine the derivative.

Question

I got the question wrong so here are my steps:

lny=ln(x^(2x))

lny=2xlnx

1/y dy/dx=2xlnx dy/dx

dy/dx=2x (1/lnx)y

dy/dx=(2x/lnx)y

Answer 1

Everthing is correct up to this step.

1/y dy/dx=2xlnx dy/dx

You are supposed to take the derivative with respect to x of both sides.

d/dx(lny) = d/dx(2xlnx)

The left side you got right.

1/y *y' or 1/y dy/dx

The right side you need to take the derivative of 2x*lnx

Use the product rule.

2*lnx + 2x * 1/x

So in total you should have written.

1/y dy/dx = 2lnx + 2

Multiply both sides by y.

dy/dx = (2lnx + 2)y

Since y = x^2x Substituted that back in.

dy/dx = (2lnx + 2)*x^2x

This is the correct answer.