- 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

One of the nifty tricks we can do with the chain rule is reconstruct the quotient rule. To find the derivative of

first rewrite the function like this:

*h*(*x*) = *f*(*x*) × (*g*(*x*))^{{-1}}.

Now that *h* is written as a product, we can use the product rule to find its derivative:

*h'*(*x*) = *f'*(*x*)(*g*(*x*))^{{-1}} + *f*(*x*)((*g*(*x*))^{{-1}})'.

Now there are two big steps left. First we'll use the chain rule to find this derivative:

*h'*(*x*) = *f'*(*x*)(*g*(*x*))^{{-1}} + *f*(*x*){ ((*g*(*x*))^{{-1}})'}.

Then we'll simplify the formula we got using the product rule until it magically turns into the quotient rule.

Chain Rule:

To finish applying the product rule, we need to know

((*g*(*x*))^{{-1}})'

In other words, we need to know the derivative of the nested function

(*g*(*x*))^{{-1}}

Do our chain rule stuff. The outside function is

(□)^{{-1}},

and its derivative is

-(□)^{{-2}}.

The inside function is

{ *g*(*x*)},

and its derivative is

*g'*(*x*).

Now we can use the chain rule:

((*g*(*x*))^{{-1}})' = -({ *g*(*x*)})^{{-2}} × *g'*(*x*)

Since (*g*(*x*))^{{-2}} is the same thing as , we can rewrite this as

Simplifying:

Returning to the product rule,

We can do some great simplifying here. Since (*g*(*x*))^{{-1}} is the same thing as , we can rewrite this as

Now we can put the fractions over a common denominator and combine them:

Ta-daa! This is the quotient rule.