Final answer:
To find h'(2) for the given functions, we can apply the rules of calculus. For a constant multiple of a function, multiply the derivative by the constant. For the sum and difference of functions, add or subtract their derivatives. Plugging in the given values, we find h'(2) for each function.
Step-by-step explanation:
a. To find h'(2) for the function h(x) = 4f(x), we can use the constant multiple rule for differentiation. The derivative of h(x) = k*f(x), where k is a constant, is given by h'(x) = k*f'(x). Substituting k=4 and x=2, we have h'(2) = 4*f'(2) = 4*3 = 12.
b. To find h'(2) for the function h(x) = 5g(x), we can apply the constant multiple rule again. The derivative of h(x) = k*g(x), where k is a constant, is given by h'(x) = k*g'(x). Plugging in k=5 and x=2, we get h'(2) = 5*g'(2) = 5*(-1) = -5.
c. To find h'(2) for the function h(x) = f(x) + g(x), we can use the sum rule for differentiation. The derivative of h(x) = f(x) + g(x) is h'(x) = f'(x) + g'(x). Substituting x=2, we have h'(2) = f'(2) + g'(2) = 3 + (-1) = 2.
d. To find h'(2) for the function h(x) = g(x) - f(x), we can apply the difference rule for differentiation. The derivative of h(x) = g(x) - f(x) is h'(x) = g'(x) - f'(x). Plugging in x=2, we get h'(2) = g'(2) - f'(2) = -1 - 3 = -4.