# Derivative of a Function

A Derivative of a Function is a real-valued function that reports the i-th slope of the tangent for a real variable function (a differentiable function).

**AKA:**Derivative.**Context:**- It can map the changes in the Function Domain of a Differentiable Function to the changes in the Function Range.
- It can range from being a Complete Derivative to being a Partial Derivative.
- It can be mapped to a Derivative Function Value.
- It can be the required output of a Derivative Function Identification Task.
- …

**Example(s):**- [math]\displaystyle{ f'(x) = (1/4)x^{-3/4}, }[/math]
- [math]\displaystyle{ \frac{d}{dx}e^x = e^x. }[/math]
- [math]\displaystyle{ \frac{d}{dx}a^x = \ln(a)a^x. }[/math]

**See:**Calculus, Velocity, Limit (Mathematics), Linear Approximation, Slope, Tangent, Differential of a Function.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/derivative Retrieved:2015-1-11.
- The
**derivative**of a function of a real variable measures the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable). It is a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced. The derivative measures the*instantaneous*rate of change of the function, as distinct from its*average*rate of change, and is defined as the limit of the average rate of change in the function as the length of the interval on which the average is computed tends to zero.The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. In fact, the derivative at a point of a function of a single variable is the slope of the tangent line to the graph of the function at that point.

The notion of derivative may be generalized to functions of several real variables. The generalized derivative is a linear map called the differential. Its matrix representation is the Jacobian matrix, which reduces to the gradient vector in the case of real-valued function of several variables.

The process of finding a derivative is called differentiation. The reverse process is called

*antidifferentiation*. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.^{[1]}

- The

- ↑ Differential calculus, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.