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 =

\(\begin{array}{l}q_{0}\end{array} \)
or f ‘ (
\(\begin{array}{l}q_{0}\end{array} \)
) represents the rate of change of p with respect to q at q =
\(\begin{array}{l}q_{0}\end{array} \)
.

Students can refer to the short notes and MCQ questions along with separate solution pdf of this chapter for quick revision from the links below:

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,

\(\begin{array}{l}\frac{\partial q}{\partial p} = \frac{\partial q}{\partial r}/\frac{\partial p}{\partial r}, if \frac{\partial p}{\partial r}\neq 0\end{array} \)

Increasing and Decreasing Function

A Function p can be said increasing when:

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

A function is said to be decreasing when:

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

Constant

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

\(\begin{array}{l} x \epsilon (a,b)\end{array} \)
and q is a constant

Equation of Tangent

The equation of a tangent is given at a point

\(\begin{array}{l}(x_{0}, y_{0})\end{array} \)
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
    \(\begin{array}{l}y^{2}=4x\end{array} \)
    .
  2. For the curve
    \(\begin{array}{l}2y+x^{2}=3\end{array} \)
    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) =
    \(\begin{array}{l}x^{2}\end{array} \)
    + ax + 1 is increasing On [1, 2]?

Also Read:

Frequently asked Questions on CBSE Class 12 Maths Notes Chapter 6: Application of Derivations

What is the ‘Chain Rule’?

The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x).

What are the uses of ‘Derivations’?

1. Determine velocity of a body 2. Solve distance and displacement problems 3. Change in population size

What is ‘Critical point’?

Critical points are places where the derivative of a function is either zero or undefined. These critical points are places on the graph where the slope of the function is zero.

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