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. The d's are not variable, and therefore cannot be cancelled out.):


Definition of a derivative[change | change source]

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[change | change source]

Linear functions[change | change source]

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[change | change source]

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[change | change source]

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[change | change source]

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

Example 2[change | change source]

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


 \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[change | change source]

The derivative of logarithms is the reciprocal:

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

Take, for example, \frac{d}{dx}\log\left(\frac{5}{x}\right). 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[change | change source]

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[change | change source]

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[change | change source]

Other websites[change | change source]