*The Derivative tells us the slope of a function at any point.*
There are **rules** we can follow to find many derivatives. For example:

- The slope of a
**constant** value (like 3) is always 0
- The slope of a
**line** like 2x is 2, or 3x is 3 etc
- and so on.

Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ’ means "Derivative of".

Common Functions | Function | Derivative |

Constant | c | 0 |

Line | x | 1 |

| ax | a |

Square | x^{2} | 2x |

Square Root | √x | (½)x^{-½} |

Exponential | e^{x} | e^{x} |

| a^{x} | ln(a) a^{x} |

Logarithms | ln(x) | 1/x |

| log_{a}(x) | 1 / (x ln(a)) |

Trigonometry (x is in radians) | sin(x) | cos(x) |

| cos(x) | −sin(x) |

| tan(x) | sec^{2}(x) |

Inverse Trigonometry | sin^{-1}(x) | 1/√(1−x^{2}) |

| cos^{-1}(x) | −1/√(1−x^{2}) |

| tan^{-1}(x) | 1/(1+x^{2}) |

| | |

Rules | Function | Derivative |

Multiplication by constant | cf | cf’ |

Power Rule | x^{n} | nx^{n−1} |

Sum Rule | f + g | f’ + g’ |

Difference Rule | f - g | f’ − g’ |

Product Rule | fg | f g’ + f’ g |

Quotient Rule | f/g | (f’ g − g’ f )/g^{2} |

Reciprocal Rule | 1/f | −f’/f^{2} |

| | |

Chain Rule
(as "Composition of Functions") | f Âº g | (f’ Âº g) × g’ |

Chain Rule (using ’ ) | f(g(x)) | f’(g(x))g’(x) |

Chain Rule (using ddx ) | dydx = dydududx |

"The derivative of" is also written ddx

So ddxsin(x) and sin(x)’ are the same thing, just written differently

##
Examples

###
Example: what is the derivative of sin(x) ?

From the table above it is listed as being **cos(x)**

It can be written as:

sin(x) = cos(x)

Or:

sin(x)’ = cos(x)

###
Power Rule

###
Example: What is x^{3} ?

The question is asking "what is the derivative of x^{3}?"

We can use the Power Rule, where n=3:

x

^{n} = nx

^{n−1}
x

^{3} = 3x

^{3−1} =

**3x**^{2}
###
Example: What is (1/x) ?

1/x is also **x**^{-1}

We can use the Power Rule, where n = −1:

x

^{n} = nx

^{n−1}
x

^{−1} = −1x

^{−1−1} =

**−x**^{−2}
###
Multiplication by constant

###
Example: What is 5x^{3 }?

the derivative of cf = cf’

the derivative of 5f = 5f’

We know (from the Power Rule):

x

^{3} = 3x

^{3−1} = 3x

^{2}
So:

5x

^{3} = 5

x

^{3} = 5 × 3x

^{2} =

**15x**^{2}
###
Sum Rule

###
Example: What is the derivative of x^{2}+x^{3 }?

The Sum Rule says:

the derivative of f + g = f’ + g’

So we can work out each derivative separately and then add them.

Using the Power Rule:

- x
^{2} = 2x
- x
^{3} = 3x^{2}

And so:

the derivative of x^{2} + x^{3} = **2x + 3x**^{2}

###
Difference Rule

It doesn't have to be **x**, we can differentiate with respect to, for example, **v**:

###
Example: What is (v^{3}−v^{4}) ?

The Difference Rule says

the derivative of f − g = f’ − g’

So we can work out each derivative separately and then subtract them.

Using the Power Rule:

- v
^{3} = 3v^{2}
- v
^{4} = 4v^{3}

And so:

the derivative of v^{3} − v^{4} = **3v**^{2} − 4v^{3}

###

###
Sum, Difference, Constant Multiplication And Power Rules

###
Example: What is (5z^{2} + z^{3} − 7z^{4}) ?

Using the Power Rule:

- z
^{2} = 2z
- z
^{3} = 3z^{2}
- z
^{4} = 4z^{3}

And so:

(5z

^{2} + z

^{3} − 7z

^{4}) = 5 × 2z + 3z

^{2} − 7 × 4z

^{3} =

**10z + 3z**^{2} − 28z^{3}

###
Product Rule

###
Example: What is the derivative of cos(x)sin(x) ?

The Product Rule says:

the derivative of fg = f g’ + f’ g

In our case:

We know (from the table above):

- cos(x) = −sin(x)
- sin(x) = cos(x)

So:

the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)

= **cos**^{2}(x) − sin^{2}(x)

###

###
Reciprocal Rule

###
Example: What is (1/x) ?

The Reciprocal Rule says:

the derivative of 1/f = −f’/f^{2}

**With f(x)= x, we know that f’(x) = 1**

So:

the derivative of 1/x = −1/x^{2}

Which is the same result we got above using the Power Rule.

###
Chain Rule

###
Example: What is ddxsin(x^{2}) ?

**sin(x**^{2}) is made up of **sin()** and **x**^{2}:

- f(g) = sin(g)
- g(x) = x
^{2}

The Chain Rule says:

the derivative of f(g(x)) = f'(g(x))g'(x)

The individual derivatives are:

- f'(g) = cos(g)
- g'(x) = 2x

So:

ddxsin(x^{2}) = cos(g(x)) (2x)

= 2x cos(x^{2})

Another way of writing the Chain Rule is: dydx = dydududx

Let's do the previous example again using that formula:

###
Example: What is ddxsin(x^{2}) ?

dydx = dydududx

Have u = x^{2}, so y = sin(u):

ddx sin(x^{2}) = ddusin(u)ddxx^{2}

Differentiate each:

ddx sin(x^{2}) = cos(u) (2x)

Substitue back u = x^{2} and simplify:

ddx sin(x^{2}) = 2x cos(x^{2})

Same result as before (thank goodness!)

Another couple of examples of the Chain Rule:

###
Example: What is (1/cos(x)) ?

**1/cos(x)** is made up of **1/g** and **cos()**:

The Chain Rule says:

the derivative of f(g(x)) = f’(g(x))g’(x)

The individual derivatives are:

- f'(g) = −1/(g
^{2})
- g'(x) = −sin(x)

So:

**(1/cos(x))’ = −1/(g(x))**^{2} × −sin(x)

**= sin****(x)**/cos^{2}(x)

Note: **sin****(x)**/cos^{2}(x) is also **tan****(x)**/cos(x), or many other forms.

###
Example: What is (5x−2)^{3} ?

The Chain Rule says:

the derivative of f(g(x)) = f’(g(x))g’(x)

**(5x-2)**^{3} is made up of **g**^{3} and **5x-2**:

The individual derivatives are:

- f'(g) = 3g
^{2} (by the Power Rule)
- g'(x) = 5

So:

**(5x−2)**^{3} = 3g(x)^{2} × 5 = 15(5x−2)^{2}