Question

Given two functions, f(x) and g(x), and the following information:

f(3) = 5
g(3) = 5
f'(3) = -3
g'(3) = 2
Calculate the values of the following derivatives.

Answer 1

1. f(x) + g(x) is differentiable at x = 3 and its derivative is f'(x) + g'(x), which equals -3 + 2 = -1.

2. The product function f(x)g(x) is differentiable at x = 3 and its derivative is f(x)g'(x) + f'(x)g(x), which equals 5(-3) + 5(2) = -15.

Step-by-step explanation:

1. To find the derivative of a sum of two functions, we use the sum rule of differentiation. At x = 3, both functions have the same function values and their derivatives have opposite signs. When adding the derivatives, we get a negative value for the derivative of the sum.

2. To find the derivative of a product of two functions, we use the product rule of differentiation. At x = 3, both functions have non-zero values, so we can apply the product rule. The derivative of the first function multiplied by the derivative of the second function, plus the derivative of the first function multiplied by the second function, gives us the derivative of their product.

In this case, we get a negative value for the derivative of the product because one function has a negative derivative and the other has a positive value at x = 3.