Suppose f(x)= x^5 - 10 and let h(x) be the inverse of f. Find h'(22).
f(x) = x^5 - 10
inverse is x = y^5 - 10
y^5 = x+10
y = (x+10)^(1/5)
h(x) = (x+10)^(1/5)
h'(x) = (1/5)(x+10^(-4/5)
h'(22) = (1/5)(32)^(-4/5)
= (1/5)(2)^-4
= (1/5)(1/16)
= 1/80
Thank you that helps a lot!
If f(a) = b then h'(b) = 1/f'(a)
f(2) = 22
f'(2) = 80
h'(22) = 1/f'(2) = 1/80
To find h'(22), we need to find the derivative of the inverse function h(x) at the point x = 22.
Step 1: Find the inverse function h(x)
To find the inverse function of f(x), we need to interchange the variables x and y and solve for y.
So, let's rewrite the equation f(x) = x^5 - 10 as y = x^5 - 10.
Step 2: Interchange x and y
Interchanging x and y, we get x = y^5 - 10.
Step 3: Solve for y
To solve for y, we need to isolate the variable y.
Adding 10 to both sides of the equation, we get x + 10 = y^5.
Step 4: Take the fifth root of both sides
Taking the fifth root of both sides, we get the inverse function y = ∛(x + 10).
Step 5: Differentiate the inverse function
To find h'(x), the derivative of the inverse function h(x), we differentiate the inverse function y = ∛(x + 10) with respect to x.
Using the chain rule, we have:
h'(x) = [(1/3)(x + 10)^(-2/3)] * (d/dx)(x + 10)
The derivative of (x + 10) with respect to x is simply 1.
Step 6: Evaluate h'(22)
Finally, we substitute x = 22 into the derived expression h'(x) and calculate the value.
h'(22) = [(1/3)(22 + 10)^(-2/3)] * 1
= [(1/3)(32)^(-2/3)]
= (1/3)(2/∛32)
= 2/(3 * ∛32)
Therefore, h'(22) = 2/(3 * ∛32).