AP Calculus AB Exam Question:

Let f(x) = (5x^3 + 2x^2)(3x - 1). Use the product rule to find the derivative of f(x), and then use the quotient rule to find the derivative of g(x) = f(x) / (x^2 + 1).

Solution:

To find the derivative of f(x) using the product rule, we differentiate each term separately and apply the formula: (uv)' = u'v + uv'.

Given: f(x) = (5x^3 + 2x^2)(3x - 1)

Step 1: Identify the functions u and v.

Let u(x) = 5x^3 + 2x^2 And v(x) = 3x - 1

Step 2: Find the derivatives of u(x) and v(x).

The derivative of u(x) with respect to x, u'(x), can be found by applying the power rule for derivatives:

u'(x) = d/dx (5x^3 + 2x^2) = 15x^2 + 4x

The derivative of v(x) with respect to x, v'(x), is simply the coefficient of x in this case:

v'(x) = d/dx (3x - 1) = 3

Step 3: Apply the product rule formula.

The derivative of f(x), f'(x), is given by:

f'(x) = u'v + uv' = (15x^2 + 4x)(3x - 1) + (5x^3 + 2x^2)(3)

Step 4: Simplify the expression.

Multiplying each term gives:

f'(x) = (45x^3 - 15x^2 + 12x^2 - 4x) + (15x^3 + 6x^2) = 60x^3 + 3x^2 - 4x

Therefore, the derivative of f(x) is f'(x) = 60x^3 + 3x^2 - 4x.

Now let's move on to finding the derivative of g(x) using the quotient rule.

Given: g(x) = f(x) / (x^2 + 1)

Step 1: Identify the functions u and v.

Let u(x) = f(x) = (5x^3 + 2x^2)(3x - 1) And v(x) = (x^2 + 1)

Step 2: Find the derivatives of u(x) and v(x).

We have already found the derivative of f(x) as f'(x) = 60x^3 + 3x^2 - 4x. Now, let's find v'(x):

v'(x) = d/dx (x^2 + 1) = 2x

Step 3: Apply the quotient rule formula.

The derivative of g(x), g'(x), is given by:

g'(x) = (u'v - uv') / (v^2)

Substituting the values, we have:

g'(x) = [(60x^3 + 3x^2 - 4x)(x^2 + 1) - (5x^3 + 2x^2)(2x)] / (x^2 + 1)^2

Step 4: Simplify the expression.

Expanding and simplifying the numerator:

g'(x) = (60x^5 + 63x^4 + 3x^2 - 4x^3 - 4x) - (10x^4 + 4x^3) = 60x^5 + 53x^4 - 10x^4 + 3x^2 - 4x^3 - 4x

Simplifying the denominator:

(x^2 + 1)^2 = (x^2 + 1)(x^2 + 1) = x^4 + 2x^2 + 1

Therefore, the derivative of g(x) is:

g'(x) = (60x^5 + 53x^4 - 10x^4 + 3x^2 - 4x^3 - 4x) / (x^4 + 2x^2 + 1)

This completes the derivative calculation of g(x) using the quotient rule.