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Derivative (mathematics)

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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):


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 [math]x_0[/math] and [math] x_1[/math] becomes infinitely small (infinitesimal). In mathematical terms,

[math]f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}}[/math]

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 [math]\frac{d}{dx}(x) = 1[/math] 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. [math]x^a[/math]) behave differently than linear functions because their slope varies (because they have an exponent).

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

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

[math]f(x) = \frac{1}{x} = x^{-1}[/math]
[math]f'(x) = -1(x^{-2})[/math]
[math]f'(x) = -\frac{1}{x^2}[/math]

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

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

Exponential functions

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

Example 1

[math]\frac{d}{dx}\left( ab^{ f\left( x \right) } \right) = ab^{f(x)} \cdot f'\left(x \right) \cdot ln(b)[/math]

Example 2

Find [math] \frac{d}{dx} \left( 3\cdot2^{3{x^2}} \right)[/math].

[math] a = 3[/math]

[math] b = 2[/math]

[math] f\left( x \right) = 3x^2[/math]

[math] f'\left( x \right) = 6x [/math]


[math] \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} [/math]

Logarithmic functions

The derivative of logarithms is the reciprocal:

[math]\frac{d}{dx}log(x) = \frac{1}{x}[/math].

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

[math]\frac{d}{dx}(log(5)) - \frac{d}{dx}(log(x))[/math]

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

[math]0 - \frac{d}{dx} log(x) = -\frac{1}{x}[/math]

Trigonometric functions

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

[math]\frac{d}{dx}sin(x) = cos(x)[/math]
[math]\frac{d}{dx}cos(x) = -sin(x)[/math].

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:

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

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