## Derivative Definition:

`Measures sensitivity to change`

Δ function Δ argument.

## Limit of a Function Notation:

`The limit of a function ƒ(x) is L`

As L approaches a...

lim_{x → a} ƒ(x) = L

## Derivative Notation:

`Given a function ƒ(x)`

The derivative is ƒ'(x).

Read this as a *f prime of x*.

Derivative of y in terms of x is:

dy/dx

## Differentiable Definition:

`ƒ(x) is differentiable at x = a`

if ƒ'(a) exists for each point

in that interval

## Differentiable/Continuous Theorem:

`If ƒ(x) is differentiable at`

x = a

then ƒ(x) is continous at x = a.

## Derivative of a constant rule:

`Derivative of a constant is 0.`

ƒ(x) = 6, then ƒ'(x) = 0

## Derivative of a variable:

ƒ(x) = x, then ƒ'(x) = 1

## Derivative power rule:

`ƒ(x) = x`^{n}, then ƒ'(x) = nx^{n - 1}

ƒ(x) = x

^{2}Using the power rule with n = 2, we have

ƒ'(x) = 2x

^{2 - 1} = 2x

## Derivative Product Rule:

`Given a term u = ƒ(x) and`

v = g(x)

where

ƒ(x) and g(x) are differentiable

then we have:

d(uv)/dx = u * dv/dx + v * du/dx

This can also be written as

(ƒ(x) * g(x))' = ƒ(x) * g'(x) + g(x) * ƒ'(x)

## Derivative Quotient Rule:

`Given a term u = ƒ(x) and`

v = g(x)

where

ƒ(x) and g(x) are differentiable

then we have

(u/v)' = | u' * v + v' * u |

| v^{2} |

This can also be written as

(ƒ(x) / g(x))' = | ƒ'(x) * g(x) + g'(x) * ƒ(x) |

| g(x)^{2} |

## Trigonometric Derivatives

ƒ(x) | ƒ'(x) | Domain |
---|

sin(x) | cos(x) | -∞ < x < ∞ |

cos(x) | -sin(x) | -∞ < x < ∞ |

tan(x) | sec^{2}(x) | x ≠ π/2 + πn, n ∈ Ζ |

csc(x) | -csc(x)cot(x) | x ≠ πn, n ∈ Ζ |

sec(x) | sec(x)tan(x) | x ≠ π/2 + πn, n ∈ Ζ |

cot(x) | -csc^{2}(x) | x ≠ πn, n ∈ Ζ |

## Logarithmic Derivatives

ƒ(x) | ƒ'(x) |
---|

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

a^{x} | a^{x} * Ln(a) |

Ln(x) | 1/x |

log_{a}x | 1/x * Ln(a) |

## Derivative Calculator:

For more help, visit out derivative calculator

##### How does the Derivatives Calculator work?

Free Derivatives Calculator - This lesson walks you through the derivative definition, rules, and examples including the power rule, derivative of a constant, chain rule

### What 5 formulas are used for the Derivatives Calculator?

lim

_{x → a} ƒ(x) = L

ƒ(x) = c, then ƒ'(x) = 0

ƒ(x) = x

^{n} then ƒ'(x) = nx

^{n - 1}ƒ(x) * g(x))' = ƒ(x) * g‘(x) + g(x) * ƒ'(x)

(u/v)' = (u' * v + v' * u)/v

^{2}For more math formulas, check out our

Formula Dossier
### What 6 concepts are covered in the Derivatives Calculator?

- constant
- a value that always assumes the same value independent of how its parameters are varied
- derivative
- rate at which the value y of the function changes with respect to the change of the variable x
- function
- relation between a set of inputs and permissible outputs

ƒ(x) - limit
- the value that a function (or sequence) approaches as the input (or index) approaches some valu
- power
- how many times to use the number in a multiplication
- variable
- Alphabetic character representing a number

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