# What is the formal definition of a derivative?

May 29, 2015

The formal definition of derivative of a function $y = f \left(x\right)$ is:
$y ' = {\lim}_{\Delta x \to 0} \frac{f \left(x + \Delta x\right) - f \left(x\right)}{\Delta x}$

The meaning of this is best understood observing the following diagram:

The secant PQ represents the mean rate of change $\frac{\Delta y}{\Delta x}$ of your function in the interval between $x$ and $x + \Delta x$.

If you want the rate of change, say, at P you "move" point Q (and the secant with it) to meet point P as in:

In doing so you must reduce $\Delta x$. If $\Delta x \to 0$ you'll get the tangent in P whose inclination will give the inclination at P and thus the derivative at P.

Hope it helps!