- 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

Start with taking the derivatives of some lines.

Find the derivative of each function.

*f*(*x*) =*x*

- g(
*x*) = 3*f*(*x*)

*h*(*x*) = 3*g*(*x*)

Answer. These are all lines.

- The derivative of
*f*(*x*) =*x*is the slope of the line*f*(*x*) =*x*, which is 1. Therefore*f'*(*x*) = 1.

- Since
*f*(*x*) =*x*,

*g*(*x*) = 3*f*(*x*) = 3*x*.

This is a line with slope 3, therefore*g'*(*x*) = 3.

- Since
*g*(*x*) = 3*x*,

*h*(*x*) = 3*g*(*x*) = 3(3*x*) = 9*x.*

This is a line with slope 9, therefore*h'*(*x*) = 9.

If we multiply a function by 3, the derivative gets multiplied by 3 also. If we multiply a function by 2, the derivative gets multiplied by 2. And so on. The derivative of the function c*f*(*x*), assuming *f* is differentiable, is c*f**'*(*x*).

In symbols, if *g*(*x*) = c*f*(*x*) where *c* is a constant, then

*g*'(*x*) = c*f**'*(*x*).

In words, if we have a function *f* and multiply it by some constant *c* to find a new function, then the derivative of that new function is *c* multiplied by the derivative of *f*. This works for any differentiable function *f* and any constant *c*.

In pictures, it's easiest to see what's going on with a line. If we take a line* y *= *mx* + *b*, it looks something like this:

If we multiply the whole line by 3, the line gets stretched vertically:

Now the line is 3 times steeper. Although it's a bit harder to see the picture with curvy functions, the idea is the same. If we stretch (or shrink) the function vertically, we're also stretching (or shrinking) its derivative.

Example 1

Let |

Example 2

Let |

Exercise 1

Find the derivative of the function.

*f*(*x*) =*x*

Exercise 2

Find the derivative of the function.

*g*(*x*) = 2*f*(*x*)

Exercise 3

Find the derivative of the function.

*h*(*x*) = 2*g*(*x*)

Exercise 4

Find the derivative of the function.

*k*(*x*) = 2*h*(*x*)

Exercise 5

Find the derivative of the function.

*f*(*x*) = 5*x*^{3}

Exercise 6

Find the derivative of the function.

*f*(*x*) = 6cos*x*

Exercise 7

Find the derivative of the function.

*f*(*x*) = 10sin*x*

Exercise 8

Find the derivative of the function.

*g*(*x*) = 6e^{x}

Exercise 9

Find the derivative of the function.

*g*(*x*) = 4 × 5^{x}

Exercise 11

Find the derivative of the function.

Exercise 12

Find the derivative of the function.

*h*(*x*) = 9*ln**x*

Exercise 13

Find the derivative of the function.

Exercise 14

Find the derivative of the function.

*h*(*x*) = 4*x*^{12}