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

Here are the steps and pieces to assemble together.

(a) f(x)= √(x² – 9 sec⁻¹x)

Use the chain rule twice

f'(x) = 1/[2√(x² – 9 sec⁻¹x)] × d(x² – 9 sec⁻¹x)/dx

d(x² – 9 sec⁻¹x)/dx = 2x – 9 d(sec⁻¹x)/dx

Use d(sec⁻¹x)/dx = 1/[x√(x² – 1)]

(b) y = 12x sin⁻¹x + √(9 – x²)

Use the product rule for 12x sin⁻¹x

d(12x sin⁻¹x)/dx = 12 sin⁻¹x + 12x d(sin⁻¹x)/dx

Use the chain rule for √(9 – x²)

√(9 – x²) = 1/[2√(9 – x²)] × d(9 – x²)/dx

d(9 – x²)/dx should be easy

Use d(sin⁻¹x)/dx = 1/[√(1 – x²)]

Hope this helps! Add a comment if you have any questions.