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, which states that if we have a composite function F(G(x)), the derivative of this composite function is the product of the derivative of the outer function (F′(u)) and the derivative of the inner function (G′(x)), evaluated at G(x).
In this case, we want to find F′(G(1)).
Since we know that F(G(x)) = x^2, we can let u = G(x). Then F(u) = u^2.
To find F′(u), we differentiate F(u) with respect to u:
F′(u) = 2u.
Now, we need to evaluate this derivative at u=G(1).
Since G′(1) = 5, this means that the derivative of G(x) with respect to x, evaluated at x=1, is 5.
So, G′(1) = 5.
Now, we substitute u=G(1) into our derivative F′(u):
F′(G(1)) = 2u = 2G(1).
Thus, F′(G(1)) is equal to 2 times G(1).
Since G(1) represents the value of the inner function G(x) evaluated at x=1, and we haven't been given the specific function G(x), we cannot determine the exact value of G(1).
Therefore, F′(G(1)) = 2G(1).