determine f'(2) for f(x)=g[h(x)] given g(2)=5, g'(2)=-3, g'(-6)=-3, h(2)=-6, h'(2)=4
To determine f'(2) for the function f(x) = g[h(x)], we can use the chain rule of differentiation. The chain rule states that the derivative of a composite function is the product of the derivative of the outer function (g) and the derivative of the inner function (h), evaluated at the given point.
The chain rule formula can be expressed as:
(f(g(x)))' = g'(x) * h'(g(x))
In this case, we are interested in finding f'(2), so we will evaluate the derivative at x = 2.
Step 1: Find the derivative of the outer function g(x) at x = 2.
Given g(2) = 5 and g'(2) = -3, we already have this information. So g'(2) = -3.
Step 2: Find the derivative of the inner function h(x) at x = 2.
Given h(2) = -6 and h'(2) = 4, we already have this information. So h'(2) = 4.
Step 3: Calculate f'(2) using the chain rule.
f'(2) = g'(2) * h'(g(2))
Substituting the known values:
f'(2) = (-3) * h'(-6)
Since we don't have the value of h'(-6), we cannot determine the exact value of f'(2) without additional information.