How do you find the derivative of #y = x^(cos x)#?

1 Answer
Jan 16, 2016

Answer:

For a function that involves a variable base to a variable exponent, we usually need some form of logarithmic differentiation.

Explanation:

#y=x^cosx#

Method 1

#lny = ln(x^cosx) = cosx lnx#

Now differentiate implicitly:

#1/y dy/dx = -sinx lnx +cosx/x#

#dy = y (cosx/x-sinxlnx) = x^cosx (cosx/x-sinxlnx)#

Method 2

#y=e^ln(x^cosx) = e^(cosxlnx)#

#dy/dx = e^(cosxlnx) d/dx(cosxlnx)# (we just did this derivative above)

# = e^(cosxlnx) (cosx/x-sinxlnx) #

# = x^cosx (cosx/x-sinxlnx) #