# JEE Application of Derivatives

This page will teach you how to master JEE Applications of Derivatives . We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.

## JEE Conceptual Theory

As per JEE syllabus, the mains concepts are Derivative as Rate Measurement, Approximation, Rolle's Theorem and Lagrange's Mean Value Theorem.

### Derivative as Rate Measurement

- \(y=f(x)\) then \(\frac{dy}{dt}=f'(x) \frac{dx}{dt}\)
- Applications

### Approximation

- Differential of a function : \(df=f'(x)dx\)
- Linearisation of a function : \(f(x)=f(x_0)+f'(x_0)(x-x_0)\)
- Geometrical significance of approximation
- Errors

### Rolle's Theorem

- Rolle's theorem and its applications
- Geometrical significance

### Lagrange's Theorem

- Lagrange's theorem and its applications
- Geometrical significance

## JEE Mains Problems

A ladder \(\SI[per-mode=symbol]{10}{\meter}\) long rests against a vertical wall. If the bottom of the ladder slides away from the wall at the rate of \(\SI[per-mode=symbol]{1}{\meter\per\second}\). How fast is the top of the ladder sliding down the wall when the bottom of the ladder is \(\SI[per-mode=symbol]{5}{\meter\per\second}\) away from the wall ?

\[\begin{array}{lll} A) \, \SI[per-mode=symbol]{\frac{1}{\sqrt{3}}}{\meter\per\second} & \quad \quad \quad \quad \quad \quad \quad & B) \, -\SI[per-mode=symbol]{\frac{1}{\sqrt{3}}}{\meter\per\second} \\ C) \, \SI[per-mode=symbol]{\sqrt{3}}{\meter\per\second} & & D) \, -\SI[per-mode=symbol]{\sqrt{3}}{\meter\per\second} \end{array}\]

Concepts tested:Rate measurement

Answer:A) \(\SI[per-mode=symbol]{\frac{1}{\sqrt{3}}}{\meter\per\second}\)

Solution:We first draw the diagram for the given situation which will look like:

Now, relationship between \(x\) and \(y\) can be established by Pythagoras Theorem: \(x^2+y^2=100\). Differentiating both sides with respect to \(t\) (time), we get : \(2x \frac{dx}{dt}+2y\frac{dy}{dt}=0 \Rightarrow \frac{dy}{dt}=-\frac{x}{y}\frac{dx}{dt}\). When \(x=5,\) the Pythagoras Theorem gives \(y=5\sqrt{3}\). Now putting the values of \(x,y\) and \(\frac{dx}{dt}\), we get: \(\frac{dy}{dt}=-\frac{5}{5\sqrt{3}}(1)=\SI[per-mode=symbol]{-\frac{1}{\sqrt{3}}}{\meter\per\second}.\)

Point to be noted here is that \(\frac{dy}{dt}\) is negative means top of the ladder is sliding down the wall at the rate of \(\SI[per-mode=symbol]{\frac{1}{\sqrt{3}}}{\meter\per\second}\). So, the correct answer is \(\SI[per-mode=symbol]{\frac{1}{\sqrt{3}}}{\meter\per\second}\).

Common mistakes:

If you considered the negative sign, then it will represent the rate with which top of the ladder will be sliding upwards the wall, while the question asks sliding rate down the wall.

If you tried to insert the values before differentiating, it will lead to zero rate, which is off course, wrong.

Let \(P(x)\) be a polynomial with real coefficients. Let \(a\) and \(b\) \((a<b)\), be two consecutive zeroes of \(P(x)\). If \(a<c<b\), then find the value of \(P'(c)+10P(c)\).

\[\begin{array}{lll} A) \, 0 & \quad \quad \quad \quad \quad \quad \quad & B) \, -1 \\ C) \, 10 & & D) \, -10 \end{array}\]

Concepts tested:Applications of Rolle's theorem

Answer:A) \(0\)

Solution:Let us assume \(f(x)=e^{10x}P(x)\). Now \(f(a)=f(b)=0\) (as \(P(a)=P(b)=0\) ) Also as \(P(x)\) is a polynomial, \(f(x)\) is continuous in \([a,b]\) and differentiable in \((a,b)\). Hence Rolle's theorem can be applied to \(f(x)\). Hence there exists \(c \in (a,b)\) such that \(f'(c)=0\).

Now, \(f'(x)=e^{10x}[P'(x)+10P(x)]\)

\( \Rightarrow e^{10c}[P'(c)+10P(c)]=0\)

\( \Rightarrow P'(c)+10P(c)=0 \text{ for some } c \in (a,b)\)

The ends \(A\) and \(B\) of a rod of length of \(\sqrt{5}\) are sliding along the curve \(y=2x^2\). Let \(x_A\) and \(x_B\) be the \(x\)-coordinates of the ends. At the moment when \(A\) is at \((0,0)\) and \(B\) is at \((1,2)\). Find the value of the derivative \(\dfrac{dx_B}{dx_A}\).

\[\lambda=\left( \frac{f(5102)-f(2015)}{f'(c)} \right) \left( \frac{f^2(2015)+f^2(5102)+f(2015)f(5102)}{f^2(c)} \right)\]

Let \(f:[2015,5102] \rightarrow [0,\infty)\)be any continuous and differentiable function. Find the value of \(\lambda\), such that there exists some \(c\in [2015,5102]\) which satisfies the equation above.

The following is a problem from JEE-Mains 2014:

## JEE Advanced Problems

A man runs along a straight path at the speed of \(\SI[per-mode=symbol]{4}{\meter\per\second}\). A searchlight is located on the ground \(\SI[per-mode=symbol]{20}{\meter}\) from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is \(\SI[per-mode=symbol]{15}{\meter}\) from the point on the path closest to the searchlight ?

Concepts tested:Rate measurement

Answer:\(\SI[per-mode=symbol]{0.128}{\radian\per\second}\)

Solution:

Step 1: Drawing figure and variable selection

We draw the figure and let \(x\) be the distance from the man to the point on the path closest to the searchlight and \(\theta\) be the distance from the man to the point on the path closest to the searchlight and perpendicular to the path. The figure will look like:Step 2: Establishing relation between \(x\) and \(\theta\) and finding the required rate

The equation that relates \(x\) and \(\theta\) can be written from the figure: \(\frac{x}{20}=\tan\theta \Rightarrow x=20 \tan\theta\). Now differentiating both sides with respect to \(t\) (time), we get : \(\frac{dx}{dt}=\SI[per-mode=symbol]{20}{\second\squared}\theta \frac{d\theta}{dt} \Rightarrow \frac{d\theta}{dt}=\frac{\cos^2\theta}{20} \frac{dx}{dt}\)

Step 3: Putting the values

We're given with \(\frac{dx}{dt}=\SI[per-mode=symbol]{4}{\meter\per\second}\) and when \(x=\SI[per-mode=symbol]{15}{\meter}\), the length of the beam becomes \(\SI[per-mode=symbol]{25}{\meter}\) using Pythagoras theorem. So, \(\cos\theta=\frac{4}{5}\).\(\Rightarrow \frac{d\theta}{dt}=\frac{1}{20} (\frac{4}{5})^2 (4)=\SI[per-mode=symbol]{0.128}{\radian\per\second}\)

Common mistakes:

- If you didn't the value of \(\frac{dx}{dt}\), it will give you linear rate, not the rate at which searchlight is rotating.

Let \(f(x)\) be a non-constant thrice differentiable function defined on real numbers such that \(f(x)=f(6-x)\) and \(f'(0)=0=f'(2)=f'(5)\). Find the minimum number of values of \(p \in [0,6]\) which satisfy the equation \[(f''(p))^2+f'(p)f'''(p)=0\]
**Details and Assumptions:**

\(f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}\)

\(f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}\)

\(f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}\)

Once a couple was caught in an unending argument due to some unknown reasons. They got irritated from each other and decided to move away. They started moving with velocities \(\SI[per-mode=symbol]{\sqrt{2+\sqrt{2}}}{\meter\per\second}\) at the same time from the junction of two roads inclined to \(45^\circ\) to each other. If they travel by different roads, find the rate at which they are being separated?

**Cite as:**JEE Application of Derivatives.

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