4.2.2 The Quotient Rule

Question 1

The Quotient Rule

Answer 1

• 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

Question 2

note

Answer 2

• 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.

Question 3

Find the derivative.f(x)=(x−2)/x^2

Answer 3

f′(x)=−x+4/x^3

Question 4

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

Answer 4

y = −7x + 2

Question 5

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

Answer 5

y = 2x − 3

Question 6

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

Answer 6

f′(x)=(x+1)(2x)−(x^2+1) /(x+1)^2

Question 7

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

Answer 7

g’(x)=11x^2+8x+6( / x−3x2+2)^2

Question 8

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

Answer 8

h’(x)=−18x^2−18 / (x^3+3x−2)^2

Question 9

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

Answer 9

f′(x)=−3/x^2