Question

Let h(x)=.75x^3+2x-1 and let g(x) be the inverse function of h(x). Notice that h(2)=5

Find g’(x)

Answer 1

To find g'(x), find the inverse function of h(x), which is f(x) = x. The derivative of f(x) is 1, so g'(x) = 1.

To find g'(x), we need to find the derivative of the inverse function g(x).

Since
`h(x) = 0.75x^3 + 2x - 1`, we can find g(x) by solving the equation h(g(x)) = x.

Let y = g(x). Then, the equation becomes h(y) = x. Rearranging this equation, we get h(y) - x = 0. Considering h(g(x)) = x, we have g(x) - x = 0.

Solving g(x) - x = 0 for x, we find g(x) = x. This means that the inverse function of h(x) is f(x) = x.

Now, to find g'(x), we differentiate g(x) = x with respect to x. Since the derivative of x with respect to x is 1, we have g'(x) = 1.