Derivatives of built in-functions:

Derivatives evaluated at a particular point:

Derivative of a user-defined function:

This is equivalent to :

Derivative at a particular value:

This is equivalent to :

The second derivative:

The third derivative:

Higher-order derivatives of a user-defined function:

f' as a shorthand for Derivative[1][f]:

Similarly, f'' is a shorthand for Derivative[2][f]:

The Derivative[k] applied to a function returns a function:

This also applies to user-defined functions:

The exception is if no derivative can be computed, in which case the input is returned unchanged:

On input, as many primes as desired can be used:

On output, symbolic derivatives of third and higher orders do not use primes:

Partial derivatives with respect to different arguments:

The partial derivative with respect to the first argument:

A mixed partial evaluated at a particular value:

Partial derivatives for functions with list arguments:

The partial derivative with respect to the first element:

A mixed partial evaluated at a particular value:

Define a derivative for a function:

Define partial derivatives for a function:

This effectively defines the gradient:

Show the vector field:

Derivative with a negative integer order can do integrals:

Derivative computes derivatives with respect to the input parameter, not any particular variables:

Any evaluation point is simply substituted into the resulting derivative function:

This is different from D, which computes a partial derivative with respect to a particular variable in an expression:

The partial derivative of an unknown function is expressed in terms of Derivative:

This extends to multivariate functions:

Partial derivatives of a multivariate function are represented by a Derivative expression with a sequence of integers:

Partial derivatives of a univariate function of a list are represented by a Derivative expression with a list of integers:

Use N to find a numerical approximation to the derivative: