Application of Derivatives

Have you ever wondered how scientists predict population growth??? Its all because of derivatives!!! Derivatives are used to model the population growth graph with the help of which scientists predict the population growth. There are a lot more application of derivatives like calculating the bacteria growth, spread of an infection, speed over time, predictions on sales over time, etc.

This article is about the concept of class 12 maths application of derivatives. Applications of derivatives chapter is essential not only for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, BCECE, and more.

Application of Derivatives

Derivatives are used to find the rate of change of a quantity. Derivatives have a lot of applications like calculating the velocity, predicting sales over time, predicting bacterial growth, the spread of infection, etc.

The application of derivatives is mainly around the concepts of rate of change of quantities, approximation, tangent and normal to a curve, increasing and decreasing function and minima and maxima of the function. Before, looking in details about this topic, let us recall what is a derivative.

Rate of Change

The value obtained after differentiating a function is called the derivative.

If two related quantities are changing over time, the rates at which the quantities change are related. For example, consider a balloon with air, both the radius and the volume of the balloon increase as the air in it increases.

If a variable quantity $y$ depend on and vary with a quantity $x$, then the rate of change of $y$ with $x$ is $\frac{d y}{d x}$.

For example, the rate of change of displacement ($s$) of an object w.r.t. time ($t$) is velocity $(\mathrm{v})$.

$
v=\frac{d s}{d t}
$

Let us consider the stone being thrown at a lake causing circular waves at a speed of 4cm per second. Now let us find how fast the enclosed area increases.

The area of a circle with radius $r$ is given by $\mathrm{A}=\pi r^2$. Therefore, the rate of change of area with respect to time $t$ is

$
\frac{d \mathrm{~A}}{d t}=\frac{d}{d t}\left(\pi r^2\right)=\frac{d}{d r}\left(\pi r^2\right) \cdot \frac{d r}{d t}=2 \pi r \frac{d r}{d t}
$

As the waves move at a speed of 4cm,

$
\frac{d r}{d t}=4 \mathrm{~cm} / \mathrm{s}
$

$\frac{d \mathrm{~A}}{d t}=2 \pi(10)(4)=80 \pi$

Thus, the enclosed area is increasing at the rate of $80 \pi \mathrm{~cm}^2 / \mathrm{s}$, when $r=10 \mathrm{~cm}$.

Linear Approximation

Linear approximation is used to estimate the value of the function close to the chosen point.

Let $f:(a, b) \rightarrow \mathbb{R}$ be a differentiable function and $x_0 \in(a, b)$. We define the linear approximation $L$ of $f$ at $x_0$ by

$
L(x)=f\left(x_0\right)+f^{\prime}\left(x_0\right)\left(x-x_0\right), \quad \forall x \in(a, b)
$

For instance, Let us assume that the shape of a soap bubble is a sphere. Now, let us approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm.

The surface area of the soap bubble with radius $r$ is $S(r)=4 \pi r^2$.

$S'(r) = 8 \pi r$

Change in the surface area

$
\begin{aligned}
S(5.2)-S(5) & \approx S^{\prime}(5)(5.2-5) \\
& =8 \pi(5)(0.2)=8 \pi \mathrm{~cm}^2
\end{aligned}
$

Now, let us look into the types of error.

Absolute Error

$\Delta \mathrm{x}$ or $d x$ is called absolute error in $x$.

Relative Error

$\frac{\Delta \mathrm{x}}{\mathrm{x}}$ or $\frac{d x}{\mathrm{x}}$ is called the relative error in $x$

Percentage Error

$\frac{\Delta \mathrm{x}}{\mathrm{x}} \cdot 100$ or $\frac{d x}{\mathrm{x}} \cdot 100$ is called the percentage error in $x$.

Tangent and Normal to a Curve

A tangent is a line touching the curve at only one point without passing through it. The tangent to a curve at a point P on it is defined as the limiting position of the secant $P Q$ as the point $Q$ approaches the point $P$ provided such a limiting position exists. The slope and equation of the tangent can be found with the use of derivatives if the curve of the function is given.

Let $P\left(x_0, y_0\right)$ be a point on the continuous curve $y=f(x)$, then the slope of the tangent to the curve at point $P$ is

$
\begin{aligned}
& \left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)} \\
& \Rightarrow\left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)}=\tan \theta=\text { slope of tangent at } P
\end{aligned}
$
Where $\theta$ is the angle which the tangent at $\mathrm{P}\left(\mathrm{x}_0, \mathrm{y}_0\right)$ makes with the positive direction of the $x$-axis as shown in the figure.

• If the tangent is parallel to $x$-axis then $\theta=0^{\circ}$.

  $
  \begin{aligned}
  & \Rightarrow \quad \tan \theta=0 \\
  & \therefore \quad\left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)}=0
  \end{aligned}
  $

• If the tangent is perpendicular to $x$-axis then $\Theta=90^{\circ}$

  $
  \begin{aligned}
  & \Rightarrow \quad \tan \theta \rightarrow \infty \quad \text { or } \quad \cot \theta=0 \\
  & \therefore\left(\frac{d x}{d y}\right)_{\left(x_0, y_0\right)}=0
  \end{aligned}
  $

Equation of Tangent

Let the equation of curve be $y=f(x)$ and let point $P\left(x_0, y_0\right)$ lies on this curve.

The slope of the tangent to the curve at a point $P$ is$
\left(\frac{d y}{d x}\right)_{\left(x_0, y_0\right)}
$

Hence, the equation of the tangent at point $P$ is $
\left(y-y_0\right)=\left(\frac{d y}{d x}\right)_{\left(x 0, y_0\right)} \cdot\left(x-x_0\right)
$

Equation of Normal

The equation of normal of a curve is $\left(y-y_0\right) f^{\prime}\left(x_0\right)+\left(x-x_0\right)=0$.

Monotonicity (Increasing and decreasing functions)

Monotonicity is the behaviour of the function whether it is increasing or decreasing. A function is said to be monotonic if it is either increasing or decreasing in its entire domain. The behaviour of the function is determined using the concept of derivatives.

A function is said to be an increasing function if it increases throughout the domain. A function $f(x)$ is increasing in $[a, b]$ if $f\left(x_2\right) \geq f\left(x_1\right)$ for all $x_2>x_1$, where $x_1, x_2 \in[a, b]$. If a function is differentiable, then $\frac{d}{d x}(f(x)) \geq 0 \quad \forall x \in(a, b)$ is a increasing function while $\frac{d}{d x}(f(x)) > 0 \quad \forall x \in(a, b)$ is a strictly increasing function.

A function is said to be a decreasing function if it decreases throughout the domain. A function $f(x)$ is decreasing in the interval $[a, b]$ if $f\left(x_2\right) \leq f\left(x_1\right)$ for all $x_2>x_1$, where $x_1, x_2 \in[a, b]$. If a function is differentiable, then $\frac{d}{d x}(f(x)) \leq 0 \quad \forall x \in(a, b)$ is a decreasing function while $\frac{d}{d x}(f(x)) < 0 \quad \forall x \in(a, b)$ is a strictly decreasing function.

For example, Let us consider a function $f(x) = x$, then $f'(x) = 1> 0$. Therefore, it is an increasing function. Similarly, let us consider a function $f(x) = -x$, then $f'(x) = -1 < 0$. Therefore, it is a decreasing function.

One of the main application of derivatives is monotonicty. This topic of monotonicity has various real life applications. This is mainly used to find the increase or decrease in profit, sales etc.

Maxima and Minima

The maximum and minimum values of a curve can be determined using derivatives.

Let $f$ be a function defined on an open interval $I$. Let $f$ be continuous at a critical point $c$ in $I$. Then

(i) If $f^{\prime}(x)$ changes sign from positive to negative as $x$ increases through $c$, i.e., if $f^{\prime}(x)>0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)<0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local maxima.

(ii) If $f^{\prime}(x)$ changes sign from negative to positive as $x$ increases through $c$, i.e., if $f^{\prime}(x)<0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)>0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local minima.

Let $y=f(x)$ be a real function defined at $x=a$. Then the function $f(x)$ is said to have a maximum value at $x=a$ if $f(x) \leq f(a) \quad \forall a \in$ R .

And also the function $f(x)$ is said to have a minimum value at $x= a$, if $f(x) \geq f(a) \quad \forall a \in R$

If at $x=a$ the shape of the curve changes from increasing to decreasing or from decreasing to increasing. Then $\mathrm{x}=\mathrm{a}$ is known as the point of inflection and $f^{\prime \prime}(x)=0$ at $x=a$

These maximum and minimum values has a lot of applications in real-life problems. It can be used to determine the maximum profit, to determine the dosage of a medicine, etc.

Now, let us summarize and recall the application of derivatives formulas.

Application of Derivatives Formulas

The application of derivatives formulas includes the formulas of rate of change, approximation and errors, equation of tangent and normal to the curve, conditions for increasing and decreasing functions, and maxima and minima.

Rate of Change

If two related quantities are changing over time, the rates at which the quantities change are related. If a variable quantity $y$ depend on and vary with a quantity $x$, then the rate of change of $y$ with $x$ is $\frac{d y}{d x} = \frac{\Delta y}{\Delta x}$.

Linear Approximation and Error

Linear approximation is used to estimate the value of the function close to the chosen point.

Let $f:(a, b) \rightarrow \mathbb{R}$ be a differentiable function and $x_0 \in(a, b)$. We define the linear approximation $L$ of $f$ at $x_0$ by

$
L(x)=f\left(x_0\right)+f^{\prime}\left(x_0\right)\left(x-x_0\right), \quad \forall x \in(a, b)
$

Tangent and Normal to a Curve

The equation of tangent to a curve at point $P$ is $
\left(y-y_0\right)=\left(\frac{d y}{d x}\right)_{\left(x 0, y_0\right)} \cdot\left(x-x_0\right)
$ where $\left(\frac{d y}{d x}\right)_{\left(x 0, y_0\right)}$ is the slope.

The equation of normal of a curve is $\left(y-y_0\right) f^{\prime}\left(x_0\right)+\left(x-x_0\right)=0$.

Monotonicity (Increasing and decreasing functions)

A function $f(x)$ is increasing in $[a, b]$ if $f\left(x_2\right) \geq f\left(x_1\right)$ for all $x_2>x_1$, where $x_1, x_2 \in[a, b]$. If a function is differentiable, then $\frac{d}{d x}(f(x)) \geq 0 \quad \forall x \in(a, b)$ is a increasing function while $\frac{d}{d x}(f(x)) > 0 \quad \forall x \in(a, b)$ is a strictly increasing function.

A function $f(x)$ is decreasing in the interval $[a, b]$ if $f\left(x_2\right) \leq f\left(x_1\right)$ for all $x_2>x_1$, where $x_1, x_2 \in[a, b]$. If a function is differentiable, then $\frac{d}{d x}(f(x)) \leq 0 \quad \forall x \in(a, b)$ is a decreasing function while $\frac{d}{d x}(f(x)) < 0 \quad \forall x \in(a, b)$ is a strictly decreasing function.

Maxima and Minima

If $f^{\prime}(x)$ changes sign from positive to negative as $x$ increases through $c$, i.e., if $f^{\prime}(x)>0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)<0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local maxima.

If $f^{\prime}(x)$ changes sign from negative to positive as $x$ increases through $c$, i.e., if $f^{\prime}(x)<0$ at every point sufficiently close to and to the left of $c$, and $f^{\prime}(x)>0$ at every point sufficiently close to and to the right of $c$, then $c$ is a point of local minima.

If at $x=a$ the shape of the curve changes from increasing to decreasing or from decreasing to increasing. Then $\mathrm{x}=\mathrm{a}$ is known as the point of inflection and $f^{\prime \prime}(x)=0$ at $x=a$

Importance of Application of Derivatives Class 12

Application of Derivatives have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

How to Study Application of Derivatives Class 12?

Start preparing by understanding and practicing the concept of derivatives. Try to be clear on concepts like rate of change, approximation, tangent and normal of a curve, monotonicity and maxma and minima. Practice many problems from each topic for better understanding. Remember to revise the application of derivatives formulas again and again.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

NCERT Notes Subject wise link:

• NCERT Notes for class 12 Physics.
• NCERT Notes for class 12 Chemistry.
• NCERT Notes for class 12 Mathematics.
• NCERT Notes for class 12 Biology

Important Books for Application of Derivatives Class 12

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Amit M Aggarwal's differential and integral calculus or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Applications of Derivatives are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.

NCERT Exemplar Solutions Subject wise link:

• NCERT exemplar solutions for class 12 Physics.
• NCERT exemplar solutions for class 12 Chemistry.
• NCERT exemplar solutions for class 12 Mathematics.
• NCERT exemplar solutions for class 12 Biology.