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 = \(q_{0}\).

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]?

Also Read:


Practise This Question

A matrix resulted from elementary row and column transformation is not equivalent to the original matrix as it happens for just row or just column transformations. Given Equivalent means the matrices can be transformed into one another by a combination of elementary row and column operations.