Thank you Reiny for replying.I know names have changed. Perhaps I should put I need to differentiate the function f(x)=(x^2-6x+23)^3/2 by the chain rule.
To differentiate the function f(x) = (x^2 - 6x + 23)^(3/2) using the chain rule, follow these steps:
Step 1: Identify the inner function.
The inner function in this case is g(x) = x^2 - 6x + 23.
Step 2: Compute g'(x), the derivative of the inner function.
To find the derivative of g(x), apply the power rule and the derivative of the linear term:
g'(x) = 2x - 6.
Step 3: Identify the outer function.
The outer function in this case is h(x) = u^(3/2), where u is the inner function g(x).
Step 4: Compute h'(x), the derivative of the outer function.
Consider h(x) = u^(3/2) as h(u) = u^(3/2). We can rewrite h(u) in terms of its power rule as h(u) = u^(3/2) = u^(1/2) * u^(1/2) * u.
Applying the power rule, the derivative of h(u) with respect to u is:
h'(u) = (1/2)u^(-1/2) * (1/2)u^(-1/2) * 1
= (1/4)u^(-1/2) * u^(-1/2)
= (1/4)(u^(-1/2 + -1/2))
= (1/4)(u^(-1)).
Step 5: Substitute g(x) for u in h'(u).
Substituting g(x) = x^2 - 6x + 23 for u, we have:
h'(x) = (1/4)(g(x))^(-1).
Step 6: Multiply h'(x) by g'(x) to find the derivative of f(x).
Now, the derivative of f(x) is given by the chain rule:
f'(x) = h'(x) * g'(x)
= (1/4)(g(x))^(-1) * (2x - 6)
= (2x - 6)/(4(g(x))).
In summary, the derivative of f(x) = (x^2 - 6x + 23)^(3/2) with respect to x, using the chain rule, is (2x - 6)/(4(x^2 - 6x + 23)^(1/2)).