AP Calculus AB Exam Question:

Let f(x) = 4x^3 + 2x^2 - 5x + 1 and g(x) = sin(x).

Find the derivative of the composite function h(x) = f(g(x)) using the chain rule.

Step-by-Step Solution:

To find the derivative of the composite function h(x) = f(g(x)), we use the chain rule, which states that if we have a composition of two functions u(v(x)), then the derivative of u(v(x)) is given by u'(v(x)) * v'(x).

Given that f(x) = 4x^3 + 2x^2 - 5x + 1 and g(x) = sin(x), we first find the derivative of f(x) using the power rule:

f'(x) = d/dx (4x^3 + 2x^2 - 5x + 1) = 12x^2 + 4x - 5

Next, we find the derivative of g(x) using the chain rule:

g'(x) = d/dx (sin(x)) = cos(x)

Now, applying the chain rule, we can find the derivative of h(x) = f(g(x)):

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

Substituting f'(g(x)) = 12(g(x))^2 + 4(g(x)) - 5 and g'(x) = cos(x):

h'(x) = (12(g(x))^2 + 4(g(x)) - 5) * cos(x)

Since g(x) = sin(x), we have:

h'(x) = (12(sin(x))^2 + 4(sin(x)) - 5) * cos(x)

Simplifying further, we have:

h'(x) = (12sin^2(x) + 4sin(x) - 5) * cos(x)

Thus, the derivative of the composite function h(x) = f(g(x)) is (12sin^2(x) + 4sin(x) - 5) * cos(x).