Calculus Homework

Question

Find the derivative of the function below.

(a) f(x)= (sqrt x^2-9^sec^-1 x)

f'(x)=

(b) y=12x sin^-1 x+(sqrt 9-x^2)

dy/dx=

Answer 1

a) f(x)=√(x²-9^arcsec(x))

First let y=9^arcsec(x)

ln(y) = arcsec(x)ln(9)

y'/y = ln(9)/(x²√(1-1/x²))

y' = yln(9)/(x²√(1-1/x²)) = 9^arcsec(x)ln(9)/(x²√(1-1/x²))

f(x) = √(x²-y)

f'(x) = (1/(2√(x²-y)))(2x-y') = (1/(2√(x²-y)))(2x-9^arcsec(x)ln(9)/(x²√(1-1/x²)))

b) y = 12xsin^-1(x)+√(9-x²)

y = 12x/(x²√(1-1/x²))+12sin^-1(x) - x/√(9-x²)