Question

Let f and g be functions that satisfy f'(2)=2 and g'(2)=-9. Find h'(2) for each function h given below:

h(x)=2f(x).

Answer 1

To find h'(2) for the function h(x) = 2f(x), we need to differentiate h(x) with respect to x and substitute the given values for f'(2) and g'(2) into the formula h'(x) = g'(f(x)) * f'(x). The value of h'(2) is 4.

Step-by-step explanation:

To find h'(2) for the function h(x) = 2f(x), we need to differentiate h(x) with respect to x and then evaluate that derivative at x = 2. Since f(x) is given as a separate function, we can differentiate it using the chain rule. The chain rule states that if we have a function g(x) = 2x, then the derivative of h(x) = g(f(x)) is equal to g'(f(x)) multiplied by f'(x). Therefore, h'(x) = g'(f(x)) * f'(x). Now, we can substitute the given values for f'(2) and g'(2) into the formula to find h'(2).

h'(x) = g'(f(x)) * f'(x) = 2 * f'(x)

Since f'(2) = 2, we have:

h'(2) = 2 * 2 = 4