Derivative (mathematics)

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

In mathematics, the derivative is a way to represent 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):

$\frac{dy}{dx}$

Definition of a derivative

The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between $x_0$ and $x_1$ becomes infinitely small (infinitesimal). In mathematical terms,

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

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, the slope of the line is 1 in all places, so $\frac{d}{dx}(x) = 1$ regardless of where the position is.

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

Power functions

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

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

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

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

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

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

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

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

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

Exponential functions

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

Example 1

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

Example 2

Find $\frac{d}{dx} \left( 3\cdot2^{3{x^2}} \right)$.

$a = 3$

$b = 2$

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

$f'\left( x \right) = 6x$

Therefore,

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

Logarithmic functions

The derivative of logarithms is the reciprocal:

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

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

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

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

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

Trigonometric functions

The cosine function is the derivative of the sine function, while the derivative of cosine is negative sine:

$\frac{d}{dx}sin(x) = cos(x)$

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

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:

$\frac{d}{dx}(3x^6 + x^2 - 6)$ can be broken up as such:

$\frac{d}{dx}(3x^6) + \frac{d}{dx}(x^2) - \frac{d}{dx}(6)$

$= 6 \cdot 3x^5 + 2x - 0$

$= 18x^5 + 2x\,$

Related pages

• Partial derivative

• Calculus

• Integral