AP Calculus AB Exam Question

Find the derivative of the following function:

f(x) = (2x^3 + 5x^2 - x + 4)^(4x)

Step-by-step Solution

To find the derivative of the function using the chain rule, we need to identify the inner and outer functions.

Let u = 2x^3 + 5x^2 - x + 4 (inner function)

Let y = u^(4x) (outer function)

Using the chain rule formula, we have:

dy/dx = dy/du * du/dx

To find dy/du, we can treat y = u^(4x) as a power of a function. Therefore, we use the power rule:

dy/du = d/du(u^(4x)) = 4x*u^(4x-1)

To find du/dx, we differentiate the inner function u = 2x^3 + 5x^2 - x + 4:

du/dx = d/dx(2x^3 + 5x^2 - x + 4) = 6x^2 + 10x - 1

Now, we can calculate the derivative dy/dx by multiplying dy/du and du/dx:

dy/dx = (4x*u^(4x-1)) * (6x^2 + 10x - 1)

Substituting u = 2x^3 + 5x^2 - x + 4, we get:

dy/dx = (4x*(2x^3 + 5x^2 - x + 4)^(4x-1)) * (6x^2 + 10x - 1)

Thus, the derivative of f(x) = (2x^3 + 5x^2 - x + 4)^(4x) is:

f'(x) = (4x*(2x^3 + 5x^2 - x + 4)^(4x-1)) * (6x^2 + 10x - 1)

This derivative represents the rate of change of the function f(x) with respect to x at any given point.