Derivative (mathematics)

A function (black) and a tangent (red). The derivative at the point is the slope of the tangent.

In mathematics, the derivative is a way to show rate of change: that is, the amount by which a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph. The derivative is often written using "dy over dx" (meaning the difference in y divided by the difference in x). The d is not a variable, and therefore cannot be cancelled out.

The derivative can be expressed as: ${\displaystyle {\frac {dy}{dx}}}$

Definition of a derivative

An animation, that gives an intuitive idea of the derivative, as the "swing" of a function change when the argument changes.

The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between ${\displaystyle x_{0}}$ and ${\displaystyle x_{1}}$ becomes infinitely small (infinitesimal). In mathematical terms,

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

That is, as the distance between the two x points (h) becomes closer to zero, the slope of the line between them comes closer to resembling a tangent line.

Derivatives of functions

Linear functions

Derivatives of linear functions (functions of the form a x + b with no quadratic or higher terms) are constant. That is, the derivative in one spot on the graph will remain the same on another.

When the dependent variable y directly takes x's value (y=x), the slope of the line is 1 in all places, so ${\displaystyle {\frac {d}{dx}}(x)=1}$ regardless of where the position is.

When y modifies x's number by adding or subtracting a constant value, the slope is still one because the change in x and y do not change if the graph is shifted up or down. That is, the slope is still 1 throughout the entire graph and its derivative is also 1.

Power functions

Power functions (e.g. ${\displaystyle x^{a}}$) behave differently than linear functions because their slope varies (because they have an exponent).

Power functions, in general, follow the rule that ${\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}}$. That is, if we give a the number 6, then ${\displaystyle {\frac {d}{dx}}x^{6}=6x^{5}}$

Another possibly not so obvious example is the function ${\displaystyle f(x)={\frac {1}{x}}}$. This is essentially the same because 1/x can be simplified to use exponents:

${\displaystyle f(x)={\frac {1}{x}}=x^{-1}}$
${\displaystyle f'(x)=-1(x^{-2})}$
${\displaystyle f'(x)=-{\frac {1}{x^{2}}}}$

In addition, roots can be changed to use fractional exponents where their derivative can be found:

${\displaystyle f(x)={\sqrt[{3}]{x^{2}}}=x^{\frac {2}{3}}}$
${\displaystyle f'(x)={\frac {2}{3}}(x^{-{\frac {1}{3}}})}$

Exponential functions

An exponential is of the form ${\displaystyle ab^{f\left(x\right)}}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are constants and ${\displaystyle f(x)}$ is a function of ${\displaystyle x}$. The difference between an exponential and a polynomial is that in a polynomial ${\displaystyle x}$ is raised to some power whereas in an exponential ${\displaystyle x}$ is in the power.

Example 1

${\displaystyle {\frac {d}{dx}}\left(ab^{f\left(x\right)}\right)=ab^{f(x)}\cdot f'\left(x\right)\cdot \ln(b)}$

Example 2

Find ${\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3{x^{2}}}\right)}$.

${\displaystyle a=3}$

${\displaystyle b=2}$

${\displaystyle f\left(x\right)=3x^{2}}$

${\displaystyle f'\left(x\right)=6x}$

Therefore,

${\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3x^{2}}\right)=3\cdot 2^{3x^{2}}\cdot 6x\cdot \ln \left(2\right)=\ln \left(2\right)\cdot 18x\cdot 2^{3x^{2}}}$

Logarithmic functions

The derivative of logarithms is the reciprocal:

${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}}$.

Take, for example, ${\displaystyle {\frac {d}{dx}}\ln \left({\frac {5}{x}}\right)}$. This can be reduced to (by the properties of logarithms):

${\displaystyle {\frac {d}{dx}}(\ln(5))-{\frac {d}{dx}}(\ln(x))}$

The logarithm of 5 is a constant, so its derivative is 0. The derivative of ln(x) is ${\displaystyle {\frac {1}{x}}}$. So,

${\displaystyle 0-{\frac {d}{dx}}\ln(x)=-{\frac {1}{x}}}$

For derivatives of logarithms not in base e like ${\displaystyle {\frac {d}{dx}}(\log _{10}(x))}$, this can be reduced to: ${\displaystyle {\frac {d}{dx}}\log _{10}(x)={\frac {d}{dx}}{\frac {\ln {x}}{\ln {10}}}={\frac {1}{\ln {10}}}{\frac {d}{dx}}\ln {x}={\frac {1}{x\ln(10)}}}$

Trigonometric functions

The cosine function is the derivative of the sine function, while the derivative of cosine is negative sine (provided that x is measured in radians):

${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}$
${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}$
${\displaystyle {\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)}$.

Properties of derivatives

Derivatives can be broken up into smaller parts where they are manageable (as they have only one of the above function characteristics), for example:

${\displaystyle {\frac {d}{dx}}(3x^{6}+x^{2}-6)}$ can be broken up as such:
${\displaystyle {\frac {d}{dx}}(3x^{6})+{\frac {d}{dx}}(x^{2})-{\frac {d}{dx}}(6)}$
${\displaystyle =6\cdot 3x^{5}+2x-0}$
${\displaystyle =18x^{5}+2x\,}$

Uses of derivatives

A function's derivative can be used to search for the maxima and minima of the function by searching for places where its slope is zero.

Derivatives are used in Newton's method which helps find zeros (roots) of a function..