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AP Calculus AB: 4.2.1 The Product Rule

Mathematics12 CardsCreated 3 months ago

This content explains the product rule, a fundamental rule in calculus used to differentiate the product of two functions. It provides the formula, a mnemonic for remembering it, and step-by-step examples. The product rule helps simplify complex derivatives and is essential when functions are multiplied together.

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product rule

The derivative of a product of two functions is not necessarily equal to the product of the two derivatives.
• The product rule states that if p(x) = f(x)g(x), where f and g are differentiable functions, then p is differentiable and
p’(x) = f(x)g’(x) + g(x)f’(x)

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Key Terms

Term
Definition

product rule

The derivative of a product of two functions is not necessarily equal to the product of the two derivatives.
• The product rule states that if p...

note

  • To find the derivative of a product of two functions you can first simplify the functions and then find the derivative of the result.

Find the derivative. f(x)=(x+1)(x^2+1)

f′(x)=3x^2+2x+1

Find the derivative. f(x)=(x+1)(x−2)

f′(x)=2x−1

Find the derivative of F(x) given that F(x)=f(x)⋅g(x),f(x)= 3x^2+1, and g(x)=√x.

F′(x)=(3x^2+1)⋅1/2x^−1/2+(x^1/2)⋅6x

Suppose f  (x) = (2x ^2 − 1) (x + 1). Which of the following lines is tangent to f and parallel to the line y = x + 2?

y = x + 1

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AP Calculus AB: 4.2.1 The Product Rule
TermDefinition

product rule

The derivative of a product of two functions is not necessarily equal to the product of the two derivatives.
• The product rule states that if p(x) = f(x)g(x), where f and g are differentiable functions, then p is differentiable and
p’(x) = f(x)g’(x) + g(x)f’(x)

note

  • To find the derivative of a product of two functions you can first simplify the functions and then find the derivative of the result.

  • At first glance, there does not seem to be any pattern to the derivative.

  • But if you decompose the product just right, you can see that the derivative is actually made up of four pieces: the original two functions and their derivatives.

  • The product rule is a shortcut for taking the derivative of a product of two functions. Remember the product rule by using the chant to the left.

  • The derivative of a product of two functions is equal to the first times the derivative of the second plus the second times the derivative of the first.

  • Expanding this function would take a lot of work. But the product rule makes finding the derivative easy.

  • You might need to do some side calculations to use the
    product rule.

  • Remember the chant! It is a simple way to remember the product rule.

Find the derivative. f(x)=(x+1)(x^2+1)

f′(x)=3x^2+2x+1

Find the derivative. f(x)=(x+1)(x−2)

f′(x)=2x−1

Find the derivative of F(x) given that F(x)=f(x)⋅g(x),f(x)= 3x^2+1, and g(x)=√x.

F′(x)=(3x^2+1)⋅1/2x^−1/2+(x^1/2)⋅6x

Suppose f  (x) = (2x ^2 − 1) (x + 1). Which of the following lines is tangent to f and parallel to the line y = x + 2?

y = x + 1

Suppose f (x) = (x^ 2 − 2x) (2x − x ^2 ). At which point is the tangent line horizontal?

(1, −1)

Find the derivative. h(x)=(2x^4+3x+7)(x^5−3x^2)

h′(x)=(2x^4+3x+7)(5x^4−6x)+(x^5−3x2)(8x^3+3)

Suppose f (x) = (x^ 2 − 2x) (3x + 2).
What is the equation of the line normal to f
(i.e., the line perpendicular to the tangent line)
at the point (1, −5)?

y+5=1/3(x−1)

Suppose f (x) = (3x − x ^2 ) (2x − x ^2 ). Find the equation of the line tangent to f at the point (1, 2).

y = x + 1

Suppose f (x) = (4x^ 3 + 3) (1 − x^ 2 ). What is the equation of the line tangent to f at the point (1, 0)?

y = −14x + 14

Find the derivative of f(x).f(x)=(x^2−1)(x^2+1)

f′(x)=4x^3