Differentiate the following: y=x^(2x)

Question

I dont understand if what is asked is for the derivative or something else.

Answer 1

They are asking for the derivative. i.e. y'.

To take the derivative here you'll want to use log rules.

Take the log of both sides:

log(y) = log(x^2x)

Using log rules, simplify the right side:

log(y) = 2x(log(x))

Take the derivative of both sides. [Note the chain rule on the left side and the product rule on the right]

y'/y = 2x(1/x) + 2(log(x))

y'/y = 2 + 2log(x)

y'/y = 2(1+log(x))

Multiply both sides by y and substitute the original equation for y:

y' = 2(1+log(x))(y)

y' = 2(1+log(x))(x^2x)

y' = 2x^2x(1+log(x))