Question

Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] y = (1 + 8x)^(1/2) (g(x), f(u)) = (1 + 7x, u^1/3)Find the derivative dy/dx. dy/dx = _______.

Answer 1

By the chain rule, you have

dy/dx = dy/du * du/dx

For the function y = f(x) = (1 + 8x)^(1/2), you're taking

y = u ^(1/2)

u = 1 + 8x

These have derivatives

dy/du = 1/2 u ^(-1/2)

du/dx = 8

which means

dy/dx = 4u ^(-1/2) = 4 (1 + 8x)^(-1/2)