## The Derivative

Derivative of a function is the limit of the ratio of the incremental change of *dependent variable* to the incremental change of *independent variable* as change of independent variable approaches zero. For the function y = f(x), the derivative is symbolized by y’ or dy/dx, where y is the dependent variable and x the independent variable.

$\displaystyle y' = \dfrac{dy}{dx} = \lim_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}$

### Interpretation of Derivative

**Slope of the Curve**

The slope of the curve *y* = *f*(*x*) at any point is identical to the derivative of the function *dy*/*dx* or *y'*.

$\text{Slope at any point, } m = y’ = \dfrac{dy}{dx}$

**Rate of Change**

The derivative of a function is identical to its rate of change. Thus, the rate of change of the volume *V* of a sphere with respect to its radius *r* is *dV*/*dr*.