Derivative (calculus)

In mathematics a derivative is measure of how functions change. Algebraic differentiation is an important part of calculus, an essential branch of mathematics. Differentiation can be used, for example, in mechanics to find the acceleration of an object from a velocity-time graph.

Essentially, differentiation is employed as a means to calculate the gradient or rate of change at a particular value for a given function, f. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph. These are a few examples of the applications of differentiation.

When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,x+h]} as $h$ tends to zero. In other words, the derivative

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}}

provided the above limit exists.

Alternative notation also commonly found is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

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.