I need help on calc 2

Question

derivative of y=x^(cos^-1(x))

Answer 1

(Just a heads up, I'm going to use arccos(x) to denote inverse cosine here.)

 

This can be a tricky sort of problem, much like differentiating x^x. Whenever there is a variable in the base AND the exponent of a function, we must use logarithmic differentiation. If that doesn't ring any bells, basically we take the log of both sides, do implicit differentiation, and solve for y'.

 

y=x^arccos(x)

 

Take ln[ ] of both sides:

ln(y) = ln[x^arccos(x)].

 

Use logarithm power rule:

ln(y) = arccos(x)*ln(x).

 

Differentiate (implicit diff on the left, product rule on the right):

(1/y)*(y') = arccos(x)*(1/x) + (-1/√[1-x²])*ln(x)

 

y'/y = arccos(x)/x - ln(x)/√[1-x²].

 

Solve for y':

y' = y*[ arccos(x)/x - ln(x)/√[1-x²] ]

 

Substitute original function in for y:

y' = [x^arccos(x)]*[ arccos(x)/x - ln(x)/√[1-x²] ]