Suppose F(G(x))=x^(2) and G′(1)=5.
Find F′(G(1)).
F′(G(1))= ___
To find F′(G(1)), we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by the product of the derivative of f(g(x)) with respect to g(x) and the derivative of g(x) with respect to x.
In this case, we have F(G(x)) = x^2. We want to find F′(G(1)), which means we need to find the derivative of F with respect to G at G(1).
Let's start by finding the derivative of F(G(x)) with respect to x:
d/dx [ F(G(x)) ] = d/dx [ x^2 ]
Since F(G(x)) = x^2, the derivative of F(G(x)) with respect to x is simply 2x.
Now, let's find the derivative of G(x) with respect to x:
d/dx [ G(x) ] = G′(x)
Given that G′(1) = 5, this means that the derivative of G(x) with respect to x is 5.
To find F′(G(1)), we need to evaluate the derivative of F with respect to G at G(1):
F′(G(1)) = 2 * G(1)
Since G(1) = 1 (from G(1) = 1), we can substitute this value:
F′(G(1)) = 2 * 1
Therefore, F′(G(1)) = 2.