# Derivative Rules

*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", and f and g are functions.

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 º g | (f’ º g) × g’ |

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

Chain Rule (using \frac{d}{dx} ) | \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} |

"The derivative of" is also written \frac{d}{dx}

So \frac{d}{dx}sin(x) and sin(x)’ both mean "The derivative of sin(x)"

## 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}**

(In other words the derivative of x^{3} is 3x^{2})

So it is simply this:

"multiply by power

then reduce power by 1"

It can also be used in cases like this:

### 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}

= \frac{−1}{x^{2}}

So we just did this:

which simplifies to **−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} = 5x^{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:

- f = cos
- g = sin

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)**

### Quotient Rule

To help you remember:

(\frac{f}{g})’ = \frac{gf’ − fg’}{g^{2}}

The derivative of "High over Low" is:

*"Low dHigh minus High dLow, over the line and square the Low"*

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

In our case:

- f = cos
- g = x

We know (from the table above):

- f' = −sin(x)
- g' = 1

So:

the derivative of \frac{cos(x)}{x} = \frac{Low dHigh minus High dLow}{over the line and square the Low}

= \frac{x(−sin(x)) − cos(x)(1)}{x^{2}}

= −\frac{xsin(x) + cos(x)}{x^{2}}

### Reciprocal Rule

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

The Reciprocal Rule says:

the derivative of \frac{1}{f} = \frac{−f’}{f^{2}}

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

So:

the derivative of \frac{1}{x} = \frac{−1}{x^{2}}

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

### Chain Rule

### Example: What is \frac{d}{dx}sin(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:

\frac{d}{dx}sin(x^{2}) = cos(g(x)) (2x)

= 2x cos(x^{2})

Another way of writing the Chain Rule is: \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

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

### Example: What is \frac{d}{dx}sin(x^{2}) ?

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

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

\frac{d}{dx} sin(x^{2}) = \frac{d}{du}sin(u)\frac{d}{dx}x^{2}

Differentiate each:

\frac{d}{dx} sin(x^{2}) = cos(u) (2x)

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

\frac{d}{dx} 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()**:

- f(g) = 1/g
- g(x) = cos(x)

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**, or many other forms.

**(x)**/cos(x)

### 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**and

^{3}**5x-2**:

- f(g) = g
^{3} - g(x) = 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}**