Question

For the given composite function, identify the inner function, u = g(x), and the outer function, y = f(u). (Use non-identity functions for f(x) and g(x).)

y = ∛(1 + 8x)
Find dy/dx.

Answer 1

The inner function is u = 1 + 8x and the outer function is y = √u. The derivative of y with respect to x is 4√(1 + 8x).

Step-by-step explanation:

The given composite function is y = √(1 + 8x). To identify the inner function and the outer function, we can rewrite the given function as y = f(g(x)), where the inner function u = g(x) is given by u = 1 + 8x and the outer function y = f(u) is given by y = √u.

To find dy/dx, we need to differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to x. Differentiating the outer function, we get dy/du = 1/2√u, and differentiating the inner function, we get du/dx = 8. Therefore, dy/dx = (dy/du)(du/dx) = (1/2√u)(8) = 4√u = 4√(1 + 8x).