Question

How do you find the derivative of `y=x^2+x^x`?

Answer 1

We know the first half. Let's try the second half, using a more foolproof way than simply trying to remember the derivative of a similar function.

Let:
`y = x^x`

`log_x(y) = log_x(x^(x)) = x`

`(lny)/(lnx) = x`

`lny = xlnx`

`1/y ((dy)/(dx)) = x*1/x + lnx*1`
(Implicit Differentiation, Product Rule)

`1/y ((dy)/(dx)) = 1 + lnx`

`((dy)/(dx))_(y = x^x) = y[1 + lnx]`

`= x^x[1 + lnx]`

Thus you have overall:

`color(blue)(((dy)/(dx))_(y = x^2 + x^x) = 2x + x^x(1+lnx))`