Question

What is the derivative of Ae^kt*cos(wt+b)?

What is the derivative of x^(x^2+1)?

Answer 1

use the product rule. d/dt uv = u'v + uv'
d/dt A e^(kt) cos(wt+b)
= A*k e^(kt) cos(wt+b) + A*e^(kt)(-w sin(wt+b))
= Ae^(kt) [k cos(wt+b) - w sin(wt+b)]

For the other, start by noting that if
y = u^v
ln y = v ln u
1/y y' = v' lnu + v * 1/u u'
y' = y/u u' + ln u y v') = v u^(v-1) u' + ln u u^v v'
This is just a combination of the power rule (d/dx u^n = n u^(n-1) u')
and the exponent rule (d/dx a^v = lna a^v v')

whew ... so
d/dx x^(x^2+1) = (x^2+1) x^(x^2) + 2x lnx x^(x^2+1)
or, if you like, x^(x^2+1) (x + 1/x + 2x lnx)