Derivative (calculus)

In mathematics a derivative is the change in the function with respect to one of it vaiables. Essentially, the derivative is 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.

The process of finding a derivative is called differentiation.

Algebraic differentiation is an important part of calculus, an essential branch of mathematics.

For a single variable real function the derivative is the equation that given the slope of the line which is tangential at that point.

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

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.

Partial derivatives

For multivalue functions $f(x_{1},x_{2},\dots ,x_{i},\dots ,x_{n})$, the partial derivative with respect to $x_{i}$ is defined as,

${\frac {\partial f(x_{1},x_{2},\dots ,x_{i},\dots ,x_{n})}{\partial x_{i}}}=\lim _{h\rightarrow 0}{\frac {f(x_{1},x_{2},\dots ,x_{i}+h,\dots ,x_{n})-f(x_{1},x_{2},\dots ,x_{i},\dots ,x_{n})}{h}}$