Differentiate the following functions

Question

a. f(x)=log₁₀(arccosx)

I started off by converting log into ln, so ln(arccosx)/ln(10). Since ln(10) is a constant, I would just differentiate ln(arccosx), right? So the answer would be (1/ln(10))*(1/(arccosx))*(-1/(√1-x²)?

b. y=tan^-1(4x²-1)

For these equations, is it as straightforward as using the chain rule and the derivative of the inverse trig equation identities? So (1/(1+x^2))*8x?

c. f(x)=2^{x^2}+arcsec(lnx)

I used the chain rule for the arcsec portion but I didn't know how to differentiate the first portion. Do I use the identity d/dx[a^f(x)]=a^f(x)*ln(a)*f'(x)? 

Answer 1

Part a. Yes and yes. But please write parentheses around the (1 + x²) in √(1 + x²) to make it clear that the radical sign covers all of the 1 + x².

Part b. I assume that your tan^-1 means "arctan". The first factor of your derivative should be

1/(1 + (4x² - 1)²), not 1/(1 + x²).

Part c. Yes. Great work!