- 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

If we didn't already know the derivative of *ln* *x*, we could figure it out using the chain rule.

We know that

*e*^{{ln x}} = *x*.

Take the derivative of each side of this equation.

The derivative of *x* is 1.

To find the derivative of *e*^{{ ln x}} we need to use the chain rule. The outside function is* e*^{{□}} and its derivative is also *e*^{{□}.}

The inside function is *ln* *x*. Since we don't yet know the derivative of *ln* *x* (at least, we're pretending we don't) we'll write its derivative as (*ln* *x*)'.

The chain rule says

(*e*^{{ ln x}})' = *e*^{{ ln x}} × (*ln* *x*)'

Since *e*^{{ln x}} = *x*, we can simplify this to

(*e*^{{ ln x}})' = *x* × (*ln* *x*)'

Now return to the equation

*e*^{{ln x}} = *x*.

The derivative of the right-hand side is 1, and the derivative of the left-hand side is *x* × (*ln* *x*)', therefore

*x* × (*ln* *x*)' = 1.

Dividing both sides by *x*, we find