Derivative

If $\Delta x$ becomes smaller and smaller, then the difference quotient approaches a number, which we can use as a measure of change. This number is called the derivative of a function.



Definition: The derivative $y'(x)$ of the function $y(x)$ in the point $x$ is the number that satisfies
$$\dfrac{y(x+\Delta x) - y(x)}{\Delta x} \rightarrow y'(x) \quad\text{if}\quad \Delta x \rightarrow 0.$$

Remark: Sometimes other notations for the derivative are used. Besides $y'(x)$ also
$$\dfrac{d}{dx}y(x) \quad\text{and}\quad \dfrac{dy}{dx}(x)$$
are often used.