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 in the modern age. 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 gradient of a function over $[x,x+h]$. If $h$ is allowed to appoach 0 then the gradient approches the gradient at the point $x$:

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

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

In classical mathematics, a function can be be differentiated using the general formula:

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

(Proving this is a worth while exercise).

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

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

Thus the derivative is a measurement of how a function changes when the values of its inputs vary. Derivatives are helpful in determining the maxima and minima of a function. For example, taking the derivative of a quadratic function will yield a linear function. The points at which this function equals zero are called critical points. Maxima and minima can occur at critical points, and can be verified to be a maximum or minimum by the second derivative test. The second derivative is used to determine the concavity, or curved shape of the graph. Where the concavity is positive, the graph curves upwards, and could contain a relative minimum. Where the concavity is negative, the graph curves downwards, and could contain a relative maximum. Where the concavity equals zero is said to be a point of inflection, meaning that it is a point where the concavity could be changing. Also, differentials have numerous applications in physics.

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,

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

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.