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 and 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[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 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. ) behave differently than linear functions because their slope varies (because they have an exponent).
Power functions, in general, follow the rule that . That is, if we give a the number 6, then
Another possibly not so obvious example is the function . 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[change | change source]
An exponential is of the form where and are constants and is a function of . The difference between an exponential and a polynomial is that in a polynomial is raised to some power whereas in an exponential is in the power.
Example 1[change | change source]
Example 2[change | change source]
Logarithmic functions[change | change source]
The derivative of logarithms is the reciprocal:
Take, for example, . 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 log(x) is . So,
Trigonometric functions[change | change source]
The cosine function is the derivative of the sine function, while the derivative of cosine is negative sine:
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:
- can be broken up as such: