 # Application Of Derivatives Class 12 Notes Chapter 6

## What are Derivatives?

Derivatives are referred to the situation when a quantity p varies with respect to another quantity q, fulfilling a condition p =f (q) and dp/dq (or f ‘ (q)) represents the change of rate of p with respect to q and dp/dq where q = $q_{0}$ or f ‘ ($q_{0}$) represents the rate of change of p with respect to q at q = $q_{0}$.

## CBSE Class 12 Maths Notes Chapter 6 Application of Derivatives – Related Links ### Chain Rule

Let us consider two variables p and q which vary with respect to the presence of another variable r, which means p = f(r) and q = g(r), then according to the chain rule,

$\frac{\partial q}{\partial p} = \frac{\partial q}{\partial r}/\frac{\partial p}{\partial r}, if \frac{\partial p}{\partial r}\neq 0$

### Increasing and Decreasing Function

A Function p can be said increasing when:

• We can see an increase on an interval (a,b) if $x_{1}<x_{2}$ in (a,b) =>$p(x_{1})<p(x_{2})$ for all $x_{1}, x_{2}\epsilon (a,b)$

A function is said to be decreasing when:

• if $x_{1}<x_{2}$ in (a,b) when: $x_{1}>x_{2}$ for all $x_{1}, x_{2}\epsilon (a,b)$

### Constant

The constant term in (a,b) if p(x)= q for all x \epsilon (a,b)\) and q is a constant

### Equation of Tangent

The equation of a tangent is given at a point $(x_{0}, y_{0})$ to the curve y =f(x) is representated by
y-yo=[dy/dx](xo,yo)(x-xo)

### Critical point

The point q inside the domain of a function f at which f ‘ (q) =0 or f cannot be differentiable is referred to as the critical point of f.

### Important Questions

1. Find out the value of m if line y=mx+1 which is a tangent to the curve $y^{2}=4x$.
2. For the curve $2y+x^{2}=3$ is ?
1. A semicircular opening is surmounted on a window which is in the form of a rectangle The total perimeter of the window is 50 m. Find the dimensions of the window to admit maximum light through the whole opening.
1. A point on the hypotenuse of a triangle is at distance p and q from the sides of the triangle.
1. For what values of a the function f given by f(x) = $x^{2}$ + ax + 1 is increasing On [1, 2]?