Derivative

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I turn away in fright and horror from this lamentable plague of functions that do not have derivatives.—C. Hermite, 1893

In calculus, the derivative of a function, where it exists, measures how a function changes as its argument changes. The (first) derivative of a differentiable function gives a measure of the rate of change of the function (i.e. the slope of its tangent) as its argument changes.

The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function 

f

at 

a

is the limit

$f'(a):=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}\,$

of the difference quotient as $\scriptstyle h\,$ approaches zero, if this limit exists. If the limit exists, then 

f

is called differentiable at 

a

. Here

f  ′(a)

is one of several common notations for the derivative.

Higher order derivatives

Second derivative

The second derivative is the derivative of the derivative of a function.

Partial derivatives

In multivariate calculus, a partial derivative of a multivariate function, where it exists, measures how a [multivariate] function changes as one of its argument changes. A (first) partial derivative of a differentiable [multivariate] function gives a measure of the rate of change of the function (i.e. the slope of a tangent) as one of its argument changes.

See also

• Nowhere differentiable functions
• Arithmetic derivative