Differentiation

Differentiation is the act of getting the rate of change of a given function, . If this was graphed, you would be getting the slope of the line tangent to the function at that point. It is one of the two main concepts of calculus and is the opposite act of integration.

When finding the derivative you must apply a set of rules to it, these rules are given in differentiation-rules.
Differentiation uses its own notation.
For finding derivatives with multiple variables see partial derivatives.
An equation that is a function of a derivative is called a differential equation.

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Derivative order
Finding a derivative

Finding Derivatives

See finding derivatives.

Derivative Order

The order of derivative is the number of time that the derivative of a function has been taken, for example if you took the derivative of it would become . This is a first order derivative. If you took the derivative of it becomes -- this is a second order derivative. See differentiation notation for information about this notation.

For example, given the function :

add a real example

Explicit and Implicit Differentiation

The difference between explicit and implicit differentiation relies of the function that you are working with. If the function ) is only reliant on one input, , then we say that the function is in an explicit form. The function is explicitly defined in terms of .

If the function is written in terms of another variable then it is said to be in an implicit form. The function is not explicitly stated but instead can be implied based off the given equation.

For more complicated implicit equations it can be difficult to get it into an explicit form, you can instead use multivariable calculus to solve the equation (see chapter 14 - sections 5-6 below).
Source and further reading: Implicit Differentiation and Multivariable Calculus Notes - Chapter 14 - Sections 5-6