Derivative (calculus)

In mathematics the derivative is the rate of change of a function with respect to one of its variables. For example, if $r(t)$ denotes the position of a particle on a line as a function of time, the derivative of $r$ with respect to the time variable $t$ is the rate of change of the particle's position with respect to time, i.e., the speed of the particle. Similarly, the derivative of the particle's speed with respect to time corresponds to the particle's velocity. Other examples of derivatives in physics abound: for example, the derivative can be used to calculate the rate of cooling from a temperature-time graph.

The process of finding a derivative is called differentiation.

Differentiation is one of two fundamental operations in calculus, the other being integration.

For a real-valued function $f(x)$ of one variable, the derivative of $f$ corresponds to the slope of the tangent line to the graph of f at x.

The roots of differentiation are profoundly linked with tangency ergo, this aspect of mathematics can first be perceived to have been developed during the time of the Ancient Greeks through the work of Greek geometers like Euclid, Sanath and Archimides.

When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over $[x,x+h]$ as $h$ tends to zero. In other words, the derivative

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

provided the above limit exists.

Alternative notation also commonly found is ${\frac {df}{dx}}=f'(x)$.

For example, a polynomial can be differentiated by taking into account the linearity of the derivative, and by using the general formula:

${\frac {d}{dx}}(x^{n})=nx^{(n-1)}$

(Proving this is a worth while exercise).

For example, if $y=3x^{2}+2x$, the derivative with respect to $x$ is

${\frac {dy}{dx}}=6x+2$

Properties of the derivative

• ${\frac {d}{dx}}(f(x)+g(x))={\frac {d}{dx}}f(x)+{\frac {d}{dx}}g(x)$

• ${\frac {d}{dx}}cf(x)=c{\frac {d}{dx}}f(x)$

• ${\frac {dy}{dx}}={\frac {dy}{dt}}\cdot {\frac {dt}{dx}}$

${\frac {dy}{dx}}$ is a proper quotient and a result,

${\frac {dy}{dx}}=f(x)\Leftrightarrow dy=f(x)dx$

is a valid operation and is much used in solving differential equations.

The differential operator has an associated eigenfunction,

${\frac {dy}{dx}}=\lambda y\Leftrightarrow y=Ce^{\lambda x}$

where e is the constant defined for this purpose.

Important differentiation rules

• Product rule
• Quotient rule
• Chain rule

Higher order derivatives

A higher order derivative is obtained by repeating derivatives,

${\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right)$

${\frac {d^{3}y}{dx^{3}}}={\frac {d}{dx}}\left({\frac {d^{2}y}{dx^{2}}}\right)={\frac {d}{dx}}\left({\frac {d}{dx}}\left({\frac {dy}{dx}}\right)\right)$

and so forth. The derivatives are termed the n-th order derivative e.g, second order derivative, third order derivative.

A common alternative notation is,

${\frac {d^{2}y}{dx^{2}}}=f''(x)$

${\frac {d^{3}y}{dx^{3}}}=f'''(x)$

${\frac {d^{n}y}{dx^{n}}}=f^{n}(x)$

The dashes are actually roman numerals: the symbol for the forth order derivative would be a slanted IV.

Picking up our earlier example; $y=3x^{2}+2x$, the second derivative with respect to $x$ is

${\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}(6x+2)=6$

${\frac {d^{3}y}{dx^{3}}}={\frac {d}{dx}}(6)=0$

All subsequent derivatives would be zero. For any polynomial of order n-1, $P_{n-1}(x)$,

${\frac {d^{n}}{dx^{n}}}P_{n-1}(x)=0$

Other than the second derivative these have no real applications^{[Citation Needed]}. A function which is differentiable infinite times is termed a smooth function.

See Also

• Partial derivatives
• Applications of derivatives