Derivative (mathematics)

In mathematics (particularly in differential calculus), the derivative is a way to show instantaneous 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 as formula_1 ("dy over dx" or "dy upon dx", meaning the difference in y divided by the difference in x). The "d" is not a variable, and therefore cannot be cancelled out. Another common notation is formula_2—the derivative of function formula_3 at point formula_4, usually read as "formula_3 prime of formula_4".
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 formula_7 and formula_8 becomes infinitely small (infinitesimal). In mathematical terms,
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 formula_10 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 formula_11 directly takes formula_4's value (formula_13), the slope of the line is 1 in all places, so formula_14 regardless of where the position is.
When formula_11 modifies formula_4's number by adding or subtracting a constant value, the slope is still 1, because the change in formula_4 and formula_11 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 (in the form of formula_19) behave differently from linear functions, because their exponent and slope vary.
Power functions, in general, follow the rule that formula_20. That is, if we give "a" the number 6, then formula_21
Another example, which is less obvious, is the function formula_22. This is essentially the same, because 1/x can be simplified to use exponents:
In addition, roots can be changed to use fractional exponents, where their derivative can be found:
Exponential functions.
An exponential function is of the form formula_28, where formula_29 and formula_30 are constants and formula_31 is a function of formula_4. The difference between an exponential and a polynomial is that in a polynomial formula_4 is raised to some power, whereas in an exponential formula_34 is in the power.
Example 1.
formula_35
Example 2.
Find formula_36.
formula_37
formula_38
formula_39
formula_40
Therefore,
formula_41
Logarithmic functions.
The derivative of logarithms is the reciprocal:
Take, for example, formula_43. This can be reduced to (by the properties of logarithms):
The logarithm of 5 is a constant, so its derivative is 0. The derivative of formula_45 is formula_46. So,
For derivatives of logarithms not in base "e", such as formula_48, this can be reduced to:
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):
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, formula_53 can be broken up as:
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 one find the zeros (roots) of a function..One can also use derivatives to determine the concavity of a function, and whether the function is increasing or decreasing.