Question

How to find the derivative of y=x^(cosx)?

Reeeally need help with this.. Have an exam in 2 hours...

Answer 1

Logarithmic differentiation would help. Write,
\[y=x^{\cos x}~~\iff~~\ln y=\ln\left(x^{\cos x}\right)\]
Log properties tell you that
\[\ln y=\cos x\ln x\]
Then differentiate both sides, which will involve chain and product rules:
\[\frac{d}{dx}\ln y=\frac{d}{dx}\cos x\ln x\\
\frac{1}{y}\frac{dy}{dx}=-\sin x\ln x+\cos x\frac{1}{x}\]
Then solve for \(\dfrac{dy}{dx}\).

Answer 2

THANK YOU THAT MAKES SO MUCH SENSE.

Answer 3

You're welcome!