- 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

The function *e* and *ln* are inverses of each other. The part we care about right now is that for any positive real number *a*,

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

If we turn this equation around, we can write any positive real number *a* as

*e*^{{ln a}}.

For example,

7 = *e*^{ln 7}

Therefore

7^{x}

is the same thing as

(*e*^{{ln 7}})^{x}

which by rules of exponents is equal to

*e*^{{(ln 7)x}}.

We can find the derivative of

*h*(*x*) = *e*^{{(ln 7)x}}

using the chain rule. The outside function is

*e*^{{□}},

whose derivative is also

*e*^{{□}},

and the inside function is

(*ln* 7)*x*,

whose derivative is the constant

(*ln* 7).

The chain rule says

*h'*(*x*) = *e*^{{(ln 7)x}} × (*ln* 7)

Turning *e*^{{(ln 7)x}} back into 7^{x}, we see that

*h'*(*x*) = 7^{x} × (*ln* 7).

This is where we find the rule for taking derivatives of exponential functions that are in other bases than *e*.