# Application Of Derivatives For Class 12

Application of derivatives includes determining the rate of change of quantities, finding the equation of tangent and normal to a curve at a point, to find the turning points on the graph of a function occurs. etc.

Rate of change of quantity-

Consider a function y = f(x), the rate of change of a function is defined as-

$\frac{dy}{dx} = f'(x)$

Further, if two variables x and y are varying to another variable, say if x = f(t), and y = g(t), then using Chain Rule, we have:

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}},$,

where $\frac{dx}{dt} \neq 0$

Increasing and Decreasing Functions:

Consider a function f, continuous in [a,b] and differentiable on the open interval (a,b), then

(i) f is increasing in [a,b] if $f'(x)>0$ for each $x \in (a,b)$

(ii) f is decreasing in [a,b] if $f'(x)< 0$ for each $x \in (a,b)$

(iii) f is constant function in [a,b], if $f'(x) = 0$ for each $x \in (a,b)$