Suppose the functions f and g and their derivatives have the following values at x = 1 and x = 2. Let h(x) = f(g(x)). Evaluate h′(1).
h'(1) = f'(g(1)) * g'(1)
Now apply your missing numbers.
-6pi
it is -6pi since your g^1(x) is -3, and your f^1(x) is 2 pi
To evaluate h′(1), we need to find the derivative of the composition function h(x) = f(g(x)) and then evaluate it at x = 1.
Since h(x) is a composition of two functions, we can use the chain rule to find its derivative. The chain rule states that if z(x) = f(g(x)), then the derivative of z(x) with respect to x is given by:
z'(x) = f'(g(x)) * g'(x).
In this case, h(x) = f(g(x)), so we can differentiate it as follows:
h'(x) = f'(g(x)) * g'(x).
To find h′(1), we need the values of f′(1) and g′(1). You mentioned that you have the values of f and g and their derivatives at x = 1 and x = 2, so we need to clarify if you have the values at both x = 1 and x = 2 for f and g, or just one of them. Let's assume you have the values at x = 1.
Using the chain rule equation, we have:
h′(x) = f'(g(x)) * g'(x).
Plugging x = 1 into this equation, we get:
h′(1) = f'(g(1)) * g'(1).
So, to evaluate h′(1), you need to know the values of f'(g(1)) and g'(1). Once you have those values, substitute them into the equation and calculate the result.