**constant**value (like 3) is always 0The slope of a

**line**like 2x is 2, or 3x is 3 etcand 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.

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Common FunctionsFunction

**Derivative**

**RulesFunctionDerivative**

Constant | c | 0 |

Line | x | 1 |

ax | a | |

Square | x2 | 2x |

Square Root | √x | (½)x-½ |

Exponential | ex | ex |

ax | ln(a) ax | |

Logarithms | ln(x) | 1/x |

loga(x) | 1 / (x ln(a)) | |

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

cos(x) | −sin(x) | |

tan(x) | sec2(x) | |

Inverse Trigonometry | sin-1(x) | 1/√(1−x2) |

cos-1(x) | −1/√(1−x2) | |

tan-1(x) | 1/(1+x2) | |

Multiplication by constant | cf | cf’ |

Power Rule | xn | nxn−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’ fg2 |

Reciprocal Rule | 1/f | −f’/f2 |

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 ddx ) | dydx = dydududx |

"The derivative of" is also written *d***dx**

So *d***dx**sin(x) and sin(x)’ both mean "The derivative of sin(x)"

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

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

It can be written as:

*d***dx**sin(x) = cos(x)

Or:

sin(x)’ = cos(x)

### Example: What is *d***dx**x3 ?

The question is asking "what is the derivative of x3 ?"

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

*d***dx**xn = nxn−1

*d***dx**x3 = 3x3−1 = **3x2**

(In other words the derivative of x3 is 3x2)

So it is simply this:

3x^2" style="width:66px; height:107px; min-width:66px;">**"multiply by powerthen reduce power by 1"**

It can also be used in cases like this:

### Example: What is *d***dx**(1/x) ?

**1/x is also x-1**

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

*d***dx**xn = nxn−1

*d***dx**x-1 = −1x-1−1

= −x-2

= *−1***x2**

So we just did this:

-x^-2" style="width:73px; height:107px; min-width:73px;">**which simplifies to −1/x2**

### Multiplication by constant

### Example: What is *d***dx**5x3 ?

the derivative of cf = cf’

the derivative of 5f = 5f’

We know (from the Power Rule):

*d***dx**x3 = 3x3−1 = 3x2

So:

*d***dx**5x3 = 5*d***dx**x3 = 5 × 3x2 = **15x2**

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### Example: What is the derivative of x2+x3 ?

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:

*d*

**dx**x2 = 2x

*d*

**dx**x3 = 3x2

And so:

the derivative of x2 + x3 = **2x + 3x2**

### Difference Rule

What we differentiate with respect to doesn"t have to be **x**, it could be anything. In this case **v**: