Derivation

Derivation is the opposite of integration. Instead of finding an area under the graph of a function, the slope of that graph can be found by using derivation. An example of an application of derivation is distance, velocity and acceleration which are time derivatives of eachother. In other words, the function for velocity and acceleration can be found by deriving the function for distance over time. This article will explain the definition of derivation.

Definition

The derivative of a function equals the slope of that function at an arbitrary point $p$. The derivative can be written as a limit as shown below:

Where $f'(x)$ is equal to the function which is the derivative of f(x), $p$ to the x-value of an arbitrary point and $dx$ a small change in x which goes to 0 in this limit.

This limit calculates the slope of a function by using the fact that the slope of a function is equal to $\frac{dy}{dx}$. This limit calculates the change in y-value in between 2 points with x-values infinitely close to eachother, which comes down to the change in y-value in 1 point ($dy$). This change in y-value is then divided by that infinitely small difference in x-value ($dx$), which results in the slope of a function in an arbitrary point ($p$).

Conclusion

The definition of a derivative most of the times is not necessary for calculating derivatives, but it gives more insight in what a derivative actually is. The definition for derivatives is now shown as well as some applications of derivatives. In the next few articles the standard rules for solving derivatives are explained in more detail.