AP Calculus AB Exam Question - Chain Rule

Find the derivative of the function f(x) = (x² + 3x + 1)⁵.

Answer: To find the derivative of the given function, we will apply the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative is given by the product of the derivative of the outer function (f'(g(x))) and the derivative of the inner function (g'(x)).

In this case, the outer function is f(x) = u⁵ and the inner function is u(x) = x² + 3x + 1. Let's apply the chain rule step by step:

Step 1: Find the derivative of the outer function, f'(u). The derivative of u⁵ with respect to u is 5u⁴.

Step 2: Find the derivative of the inner function, u'(x). The inner function is u(x) = x² + 3x + 1. Using the power rule and the sum rule, we can find the derivative of u(x) as follows: u'(x) = d/dx (x² + 3x + 1) = d/dx (x²) + d/dx (3x) + d/dx (1) = 2x + 3 + 0 = 2x + 3.

Step 3: Apply the chain rule. The chain rule states that f'(x) = f'(u) * u'(x). So, substituting the values we found in Steps 1 and 2: f'(x) = 5u⁴ * (2x + 3).

Step 4: Substitute the value of u(x) back into the expression. The inner function was u(x) = x² + 3x + 1. So, substituting u(x) = x² + 3x + 1 back into f'(x): f'(x) = 5(x² + 3x + 1)⁴ * (2x + 3).

Therefore, the derivative of the function f(x) = (x² + 3x + 1)⁵ is f'(x) = 5(x² + 3x + 1)⁴ * (2x + 3).