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AP Calculus AB: 4.2.2 The Quotient Rule

Mathematics9 CardsCreated 8 months ago

This content covers the quotient rule, a method for finding the derivative of a quotient of two differentiable functions. It provides the formula, a helpful chant for memorization, and step-by-step examples. The quotient rule is essential when dealing with functions divided by each other and is especially useful when simplification is difficult through algebra alone.

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The Quotient Rule

  • Similar to the product rule, the derivative of a quotient of two functions is not necessarily equal to the quotient of the derivatives.

  • The quotient rule states that if q(x) = f(x)/g(x), where f and g are differentiable functions, then q is differentiable except where g(x)=0 and q’(x) = g(x)f’(x)-f(x)g’(x) / [g(x)]^2

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Key Terms

Term
Definition

The Quotient Rule

  • Similar to the product rule, the derivative of a quotient of two functions is not necessarily equal to the quotient of the derivatives.

    ...

note

  • The derivative of a quotient of two functions is not necessarily equal to the quotient of the derivatives.

  • Much like the pro...

Find the derivative. f(x)=(x−2)/x^2

f′(x)=−x+4/x^3

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

y = −7x + 2

What is the equation of the line tangent to f at the point (2, 1)?
f(x)=x^2−3 / 2x−3

y = 2x − 3

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

f′(x)=(x+1)(2x)−(x^2+1) /(x+1)^2

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AP Calculus AB: 4.2.2 The Quotient Rule
TermDefinition

The Quotient Rule

  • Similar to the product rule, the derivative of a quotient of two functions is not necessarily equal to the quotient of the derivatives.

  • The quotient rule states that if q(x) = f(x)/g(x), where f and g are differentiable functions, then q is differentiable except where g(x)=0 and q’(x) = g(x)f’(x)-f(x)g’(x) / [g(x)]^2

note

  • The derivative of a quotient of two functions is not necessarily equal to the quotient of the derivatives.

  • Much like the product rule, there is a shortcut you can use to find the derivative of a quotient. The shortcut is called the quotient rule.

  • Try to remember this chant when using the quotient rule.

  • Here is the formula for the quotient rule.

  • To use the quotient rule you will need to find the derivative of the numerator and the denominator. Then just combine the different pieces according to the chant.

  • Combining terms and canceling can often simplify the result.

  • Here is another example of the quotient rule.

  • Multiply the denominator by the derivative of the numerator, subtract the numerator multiplied by the derivative of the denominator, and divide everything by the denominator squared.

Find the derivative. f(x)=(x−2)/x^2

f′(x)=−x+4/x^3

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

y = −7x + 2

What is the equation of the line tangent to f at the point (2, 1)?
f(x)=x^2−3 / 2x−3

y = 2x − 3

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

f′(x)=(x+1)(2x)−(x^2+1) /(x+1)^2

Find the derivative of: g(x)=2x^2+3x / x−3x^2+2.

g’(x)=11x^2+8x+6( / x−3x2+2)^2

Find the derivative of: h(x)=x^3+3x+4 / x^3+3x−2.

h’(x)=−18x^2−18 / (x^3+3x−2)^2

Find the derivative. f(x)=x+3/x

f′(x)=−3/x^2