- Topics At a Glance
- Derivatives of Basic Functions
- Constant Functions
- Lines
- Power Functions
- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions
- Derivatives of More Complicated Functions
- Derivative of a Constant Multiple of a Function
- Multiplication by -1
- Fractions With a Constant Denominator
- More Derivatives of Logarithms
- Derivative of a Sum (or Difference) of Functions
- Derivative of a Product of Functions
**Derivative of a Quotient of Functions**- Derivatives of Those Other Trig Functions
- Solving Derivatives
- Using the Correct Rule(s)
- Giving the Correct Answers
- Derivatives of Even More Complicated Functions
- The Chain Rule
- Re-Constructing the Quotient Rule
- Derivative of a
^{x} - Derivative of
*ln*x - Derivatives of Inverse Trigonometric Functions
- The Chain Rule in Leibniz Notation
- Patterns
- Thinking Backwards
- Implicit Differentiation
- Computing Derivatives Using Implicit Differentiation
- Using Leibniz Notation
- Using Lagrange Notation
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Division within derivatives is more complicated that the other rules we've seen so far. Make some space in the ol' memory bank for the Quotient Rule. The Quotient Rule states that the derivative of the function is

The quotient rule is more complicated than the product rule. Here are some important bits that need to be remembered.

- The numerator involves { subtraction}. Not addition, but subtraction:

- The denominator is the { square } of the original function's denominator:

- The numerator is

*f 'g*-*fg'*,

not the other way around.

One way to remember this is that we read the numerator of a fraction first, and in the quotient rule we { take the derivative of the numerator first}:

There's also a mnemonic that may be helpful. If we think of the original function as

then the numerator in the quotient rule is

"low dee-high minus high dee-low," where "dee" means "derivative." The phrase does have a nice ring to it.

"low dee-high minus high dee-low" translates to

*gf *' - *fg*',

but this is the same thing as

*f 'g* - *fg'*.

That's a lot of stuff to remember, but practice will make it easier. After we learn the Chain Rule we'll be able to re-create the Quotient Rule.

If a function is written as a fraction, it doesn't necessarily mean we need to use the quotient rule to find the derivative. If the denominator of a function is a constant, we can rewrite the function and avoid using the quotient rule.

Example 1

Find the derivative of the function |

Example 2

Find the derivative of |

Example 3

Find the derivative of the function |

Exercise 1

Find the derivative of the function. They may not all require the quotient rule.

Exercise 2

Find the derivative of the function. They may not all require the quotient rule.

Exercise 3

Find the derivative of the function. They may not all require the quotient rule.

Exercise 4

Find the derivative of the function. They may not all require the quotient rule.

Exercise 5

Find the derivative of the function. They may not all require the quotient rule.

Exercise 6

Find the derivative of the function. They may not all require the quotient rule.

Exercise 7

Find the derivative of the function. They may not all require the quotient rule.

Exercise 8

Find the derivative of the function. They may not all require the quotient rule.

*f*(*x*) = log_{x}5

Exercise 9

Find the derivative of the function. They may not all require the quotient rule.

Exercise 10

Find the derivative of the function. They may not all require the quotient rule.