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...
limx → 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) = 1Derivative power rule:
ƒ(x) = xn, then ƒ'(x) = nxn - 1ƒ(x) = x2
Using the power rule with n = 2, we have
ƒ'(x) = 2x2 - 1 = 2x
Derivative Product Rule:
Given a term u = ƒ(x) andThis can also be written as
v = g(x)
where
ƒ(x) and g(x) are differentiable
then we have:
d(uv)/dx = u * dv/dx + v * du/dx
(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 |
| v2 |
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) | sec2(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) | -csc2(x) | x ≠ πn, n ∈ Ζ |
Logarithmic Derivatives
| ƒ(x) | ƒ'(x) |
|---|---|
| ex | ex |
| ax | ax * Ln(a) |
| Ln(x) | 1/x |
| logax | 1/x * Ln(a) |
Derivative Calculator:
For more help, visit out derivative calculatorHow 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?
limx → a ƒ(x) = L
ƒ(x) = c, then ƒ'(x) = 0
ƒ(x) = xn then ƒ'(x) = nxn - 1
ƒ(x) * g(x))' = ƒ(x) * g‘(x) + g(x) * ƒ'(x)
(u/v)' = (u' * v + v' * u)/v2
For more math formulas, check out our Formula Dossier
ƒ(x) = c, then ƒ'(x) = 0
ƒ(x) = xn then ƒ'(x) = nxn - 1
ƒ(x) * g(x))' = ƒ(x) * g‘(x) + g(x) * ƒ'(x)
(u/v)' = (u' * v + v' * u)/v2
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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