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.