NCERT Maths Class 12 Chapter 6 Application of Derivatives is an important chapter. It is advised that students should revise this chapter as many times as possible. To help students do that, the best quality Class 12 Maths Chapter 6 Revision Notes by Vedantu is the answer. You can also choose to download the Class 12 Maths Notes Chapter 6 Application of Derivatives Notes PDF for offline revision purposes. Let’s start the NCERT Class 12 Maths Chapter 6 Application of Derivatives Notes by understanding what the term exactly means.
The following are the important topics that are covered under the chapter on the
Application of Derivatives:
In the table below we have provided the PDF links of all the chapters of CBSE Class 12 Maths whereby the students are required to revise the chapters by downloading the PDF.
CBSE Class 12 Maths Chapter-wise Notes
Chapter 6: Application of Derivatives Notes
It is a curated compilation of relevant online resources that complement and expand upon the content covered in a specific chapter. Explore these links to access additional readings, explanatory videos, practice exercises, and other valuable materials that enhance your understanding of the chapter's subject matter.
Application of Derivatives Related Study Materials
Find a curated selection of study resources for Class 12 subjects, helping students prepare effectively and excel in their academic pursuits.
Important Class 12 Related Links
Competitive Exams after 12th Science1. If $V=\frac\pi r^$ , at what rate in cubic units is V increasing, when r = 10 and $\frac=0.01$ ?
Ans. Given: r=10 and $\frac=0.01$
We know that $V=\frac\pi r^$
$\Rightarrow \frac=4\pi (10)^\times0.01$
2. Find the slope of tangent to the curve $y=e^$ at the point (0, 1).
Ans. Equation of curve: $y=e^$
$\therefore \left ( \frac \right )_=2.e^=2=$ Slope of tangent to the curve.
3. Find the interval on which the function $f(x)=2x^+9x+12$ is decreasing.
Ans. We have, $f(x)=2x^+9x+12$
For f(x) to be decreasing, $f<>'(x)\leq 0$
$\Rightarrow 4x+9\leq 0\;\Rightarrow x\leq -\frac$
$\Rightarrow x\epsilon \left ( -\infty ,-\frac \right ]$.
4. Let $f:R\rightarrow R$ be defined by $f(x)=2x+cos\;x$ then prove that f(x) is an increasing function
Ans. We have, $f(x)=2x+cos\;x$
$\therefore f<>'(x)=2-sin\;x>0,\;\forall x$
hence, f(x) is an increasing function.
5. Prove that, in a sphere the rate of change of surface area is $8\pi$ times the rate of change of radius.
Ans. Let r be the radius and S be the surface area of the sphere at any time t .
$\therefore$ The rate of change of surface area is 8π times the rate of change of the radius.
6. The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. Find the rate at which the area increases, when side is 10 cm.
Ans. Let the side of triangle $\rightarrow x$ Area $\rightarrow A$ .
To find : \frac at x=10cm
We know that, Area of equilateral triangle, $A=\frac>x^$
Differentiating w.r.t. ‘t’, we get
7. If x is real, then what is the minimum value of x 2 – 8 x + 17.
Ans. We have, $f(x)=x^-8x+17$
Now, $f<>''(x)=2>0,\;\forall x\;\epsilon R$
So, x=4 is the point of minima.
Minimum value of f(x) at x=4.
8. If an error of k% is made in measuring the radius of a sphere, then find the percentage error in its volume.
Ans. Let x be the radius of the sphere and y be its volume. Then,
Hence, the error in the volume is 3k% .
9. Find the approximate value of $(33)^$.
Ans. Consider the function $y=f(x)=x^>$ .
Let: x=32, $\Delta x=1$
10. Find the equation of normal to the curve y= tan x at (0,0).
Ans. We have y= tan x
$\Rightarrow \left ( \dfrac \right )_=sec^0=1$ slope of tangent
$\Rightarrow$ Slope of normal at point (0,0) is $=\frac<\left ( \dfrac
$\therefore$ Equation of normal :
11. Find the approximate value of f(3.02), where f(x)=3x 2 +5x+3.
Ans. Let x=3 and $\Delta x=0.02$ .
\begin f(x) &= 3x^2 + 5x + 3 \\ f(3) &= 3 \cdot 3^2 + 5 \cdot 3 + 3 = 45 \\ Now, y &= f(x) \\ \Delta y &= Ay \\ \Delta y &= (6x + 5) \cdot \Delta x \\ \Delta y &= (6 \cdot 3 + 5) \cdot 0.02 = 0.46 \\ f(3.02) &= f(x + \Delta x) = y + \Delta y = 45 + 0.46 = 45.46 \end
Hence, the approximate value of f(3.02) is 45.46.
12. Find the approximate value of (1.999) 5
Ans. Let x=2 and \Delta x=-0.001
Now, $$\begin& \Delta y=\dfrac \Delta x=5 x^4 \times \Delta x=5 \times 2^4 \times[-0.001] \\ & =-80 \times 0.001=-0.080 \\ & \therefore(1.999)^5=y+\Delta y=2^5+(-0.080) \\ & =32-0.080=31.920 \end $$
13. Prove that the function $f$ given by $f(x)=\log \sin x$ is strictly increasing on $\left(0, \frac<\pi>\right)$ and strictly decreasing on $\left(\frac<\pi>, \pi\right)$.
Ans. We have, $f(x)=\log \sin x$
In interval $\left(0, \dfrac<\pi>\right), f^<\prime>(x)=\cot x>0$
$\therefore f$ is strictly increasing in $\left(0, \dfrac<\pi>\right)$
In interval $\left(\dfrac<\pi>, \pi\right), f^<\prime>(x)=\cot x
$\therefore f$ is strictly decreasing in $\left(\dfrac<\pi>, \pi\right)$.
PDF Summary - Class 12 Maths Application of Derivatives Notes (Chapter 6)
In various fields of applied mathematics, one has the quest to know the rate at which one variable is changing, with respect to another. The rate of change naturally refers to time. But we can have a rate of change with respect to other variables also.
All questions of the above type can be interpreted and represented using derivatives.
The average rate of change of a function:
The average rate of change of a function $f(x)$ with respect to $x$ over an interval $[a,a + h]$ is defined as $\dfrac<
Instantaneous rate of change of a function:
The instantaneous rate of change of a function $f$ with respect to $x$ is defined as
$f(x) = \mathop <\lim >\limits_ \dfrac<
(I)The value of the derivative at $P\left( ,> \right)$ gives the slope of the tangent to the curve at \[P\] Symbolically,
Example: $x = 0$ is a tangent to $y = >>$ at $(0,0)$ .
Example: If the equation of a curve be $ - + + 3y - = 0$ , the tangents at the origin are given by $ - = 0$ that is, $x + y = 0$ and $x - y = 0$ .
(b) Length of Subtangent $(MT) = \dfrac>>>\left( > \right)>>$
(d) Length of Subnormal $(MN) =
The differential of a function is equal to its derivative multiplied by the differential of the independent variable.
Thus if, $y = \tan x$ then $dy = xdx$
Given a point $P(a,b)$ which does not lie on the curve $y = f(x)$ , then the equation of possible tangents to the curve $y = f(x)$ , passing through $(a,b)$ can be found by solving for the point of contact $Q$ .
And equation of tangent is $y - b = \dfrac>>(x - a)$
The angle between two intersecting curves is defined as the acute angle between their tangents or the normals at the point of intersection of two curves.
Here, $\,\,>\,\,>_2>$ are the slopes of tangents at the intersection point $\left( ,> \right)$ .
Shortest distance between two non-intersecting differentiable curves is always along their common normal.
From definition of derivative, $\mathop <\lim >\limits_ \dfrac>> = \dfrac>>$
(i) $\delta x$ is known as absolute error in $x$ .
(ii) $\dfrac>$ is known as relative error in $x$ .
(iii) $\dfrac> \times 100$ is known as the percentage error in $x$ .
1.A function $f(x)$ is called an Increasing Function at a point $x$ $ = $ if in a sufficiently small neighborhood around $x = a$ . We have
Similarly Decreasing Function if
Above statements hold true irrespective of whether $f$ is non derivable or even
discontinuous at $x = a.$
2. A differentiable function is called increasing in an interval $(a,b)$ if it is increasing at every point within the interval (but not necessarily at the endpoints). A function decreasing in an interval $(a,b)$ is similarly defined.
3. A function which in a given interval is increasing or decreasing is called "Monotonic" in that interval.
4. Tests for increasing and decreasing of a function at a point :
If the derivative $
If $
Example: $f(x) = $ is increasing at every point. Note that, $\dfrac>> = 3$
Let $f(x)$ be a function of $x$ subject to the following conditions:
(i) $f(>)$ is a continuous function of $>$ in the closed interval of $a \leqslant x \leqslant b$
(ii) $
Then there exists at least one point $x = c$ such that $a < c < b$ where $
Let $f(x)$ be a function of $x$ subject to the following conditions:
(i) $f(x)$ is a continuous function of $x$ in the closed interval of $a \leqslant x \leqslant b$ .
(ii) $
Then there exists at least one point $x = c$ such that $a < c < b$ where $
Geometrically, the slope of the secant line joining the curve at $x = a\,\,>\,\,x = b$ is equal to the slope of the tangent line drawn to the curve at $x = c$ .
Note the following : Rolle's theorem is a special case of LMVT since
This interpretation of the theorem justifies the name "Mean Value" for the theorem.
(c) Application of rolles theorem for isolating the real roots of an equation $f(x) = 0$
Suppose $a\,\& \,\;b$ are two real numbers such that ;
(i) $f(x)>\& $ its first derivative $
(ii) $f(a)$ and $f(b)$ have opposite signs
(iii) \[
Then there is one and only one real root of the equation $f(x) = 0$ between $a\,\& \,b$ .
1.A function $f(x)$ is said to have a local maximum at $x = a$ if $f(a)$ is greater than every other value assumed by $f(x)$ in the immediate neighbourhood of $x = a$ .
$\left. > f(\;a + h)> \\ f(\;a - h)> \end> \right] $ \Rightarrow x = a$ gives maxima for a sufficiently small positive $h$ .
Similarly, a function $f(x)$ is said to have a local minimum value at $x = b$ if $f(\;>)$ is least than every other value assumed by $f(x)$ in the immediate neighborhood at $x = b$ . Symbolically if
$\left. > \\ \end> \right] \Rightarrow x = b$ gives minima for a sufficiently small positive $h$ .
(image will be uploaded)
2. A necessary condition for maxima and minima
If $f(x)$ is a maxima or minima at $x = c$ and if $
(i) $\dfrac>> = 0$ , if it exists;
(ii) or it fails to exist
3.Sufficient condition for extreme values First Derivative Test
$\left. > <
Where $h$ is a sufficiently small positive quantity.
$\left. > <
Where $h$ is a sufficiently small positive quantity.
Note : $ -
4.Use of second order derivative in ascertaining the maxima or minima
(a) $f(c)$ is a minima of the function $f$ , if $
5. Summary-working rule
The usual tests are :
$ \Rightarrow y$ is minima.
$ \Rightarrow y$ is maxima.
If $\dfracy>>>> = 0$ when $\dfrac>> = 0$ , the test fails.
But if $\dfrac>>$ changes sign from negative to zero to positive as $x$ advances through $$ , there is a minimum. If $\dfrac>>$ does not change sign, neither a maxima nor a minima. Such points are called INFLECTION POINTS.
(In general, check at all Critical Points).
6. Useful formulae of mensuration to remember
Volume of a cuboid $ = lbh$ .
Surface area of a cuboid $ = 2(l\;b + bh + hl)$
Volume of a prism \[ = \] area of the base $ \times $ height.
Lateral surface of a prism \[ = \] perimeter of the base $ \times $ height. Total surface of a prism \[ = \] lateral surface $ + \,\,2$ area of the base
(Note that lateral surfaces of a prism are all rectangles).
Volume of a pyramid $ = \dfrac$ area of the base $ \times $ height.
Curved surface of a pyramid \[ = \]$\dfrac$ (perimeter of the base) $ \times $ slant height.
Volume of a cone $ = \dfrac\pi \;h$
Curved surface of a cylinder $ = 2\pi rh$ .
Total surface of a cylinder $ = 2\pi rh + 2\pi $
Volume of a sphere $ = \dfrac\pi $
Surface area of a sphere $ = 4\pi $
Area of a circular sector $ = \dfrac\theta $ , where $\theta $ is in radians.
7.Significance of the sign of 2nd order derivative and points of inflection
The sign of the $>>>$ order derivative determines the concavity of the curve. Such points such as $C\,\,\& \,\,E$ on the graph where the concavity of the curve changes are called the points of inflection. From the graph we find that if:
(image will be uploaded)
(i) $\dfrac>>>> > 0 \Rightarrow $ Concave upwards
At the point of inflection we find that $\dfrac>>>> = 0$ and $\dfrac>>>>$ changes sign.
Inflection points can also occur if $\dfrac>>>>$ fails to exist (but changes its sign).
For example, consider the graph of the function defined as,
$(x = c)$ and the other a point of inflection $(x = 0)$ . This implies that not every Critical Point is a point of extrema.
In earlier chapters, students must have learned how to find the derivatives of different functions, including implicit functions, trigonometric functions, and logarithmic functions. There are many applications of the derivatives of those functions. These applications lie in both mathematical concepts and real-life scenarios. Some of those applications are:
There are definitely more applications of derivatives. However, there is one important question that we still have to ask ourselves while going through the revision notes Class 12 Chapter 6. And that question is, what exactly are derivatives?
In the simplest terms, derivatives are defined as the rate of change of one quantity to another. When it comes to the terms of functions, then this rate of change of function is depicted as dy / dx = f (x) = y’.
Usually, the concept of derivatives is used in both large scale and small scale industries. This concept plays an important role when it comes to determining or bringing about the change of temperature or rate of change of sizes and shapes of an object. This last part depends on various conditions.
Before we jump to the next section of these revision notes, Class 12 Math Chapter 6, let’s look at the important applications of derivatives in more depth.
The rate of change of a quantity is the most important and general application of derivatives. For example, if one wants to check the rate of change of the volume of a cube for the decreasing side, then one can use the derivative form of dy / dx.
In this equation, dy represents the rate of change of volume of a cube. On the other hand, dx represents the change of the sides of the cube.
Derivatives are used whenever one wishes to find out whether a given function is decreasing or increasing or it remains constant. This can be done with the help of a graph. Also, if f is a function that is continuous in [p, q] and differentiable in the open interval (p, q), then:
The next thing that we will learn in these Maths Class 12 Chapter 6 Revision Notes is the application of tangent and normal to a curve. But first, let’s look at the basics. A tangent can be defined as the line that touches the curve at a point. This line doesn’t cross the curve. Also, the normal is the line that is perpendicular to the tangent.
We can also write the straight-line equation that passes through a point, which has a slope m, as:
y- y 1 = m (x - x 1 )
From this equation above, we can see that the slope of the tangent to the curve y = f (x) and at the point P (x 1 , y 1 ), it is given as dy / dx at P (x 1 , y 1 ) = f’ (x).
Hence, the equation of the tangent to the curve at P (x1, y1) can also be written as:
y - y 1 = f’ (x 1 ) (x - x 1 )
We can also write the equation of normal to the curve as:
y - y 1 = [-1 / f’ (x 1 )] (x - x 1 )
The same can also be written as:
(y - y 1 ) f’ (x 1 ) + (x - x 1 ) = 0
Another important topic that we need to look at during these NCERT Class 12 Revision Notes Maths Chapter 6 Solution is of minima and maxima. According to notes, the lowest point of a graph is known as minima. And the highest point in a graph is known as maxima.
If an individual wants to calculate the lowest and highest point of the curve in a graph or find the turning point, then a derivative function can be used.
Monotonicity is another important topic that students should learn while going through NCERT Solutions Chapter 6 Class 12 Maths Revision Notes. As described in the notes, a function can be said to be monotonic if it has either decreasing or increasing in its entire domain. For example, f (x) = ex, f (x) = nx, and f (x) = 2x + 3. On the other hand, functions that are decreasing or increasing in particular domains are known as non-monotonic. For example, f (x) = x 2 and f (x) = sin x.
Also, what about the monotonicity of a function at a point? In that case, a function is called monotonically decreasing at x = a, if f (x) satisfies the following conditions:
Before going through the Application Of Derivatives Class 12 Maths Revision Notes, students should also know how to find the approximate value or approximation. But what if you have skipped this topic earlier? In that case, readers can learn about this topic now.
If one wants to find a very small variation or change in a quantity, then he or she can use derivatives. This would provide the individual with an approximate value of it. The approximate value can also be represented by △ or delta.
Let’s assume that change in the value of x, dx = x
In that case, dy / dx = △x = x
This further means that if the change in x, dx is almost equal to x, then dy is also almost equal to y.
Another important topic that we need to discuss is the point of inflection. In case of a continuous function f (x), if f’ (x0) = 0 or f” (x0) does not exist at locations or points where f’ (x0) exists and if f” (x) changes sign when it is passing through x = x0, then x0 is known as the point of inflection.
There are also two important cases here. And these cases are:
For example, if we have to solve the equation f (x) = sin x, then we can solve it in the manner that is mentioned below.
F” (x) = sin x = 0 x = n π, n ∈ z
Did you know that derivatives are used in real life to calculate the amount of loss and profit that occurred in a business? This is done by using graphs. You can also check the variations in temperatures by using derivatives. It is also possible to find out the distance or speed covered in terms of kilometers per hour or miles per hour by using derivatives.
Apart from all of this, derivatives also play an important role in deriving various equations in the subject of physics. Derivatives also make the study of seismology possible. It can do that by allowing individuals to find the range of magnitudes of any particular earthquake.
The following is a list of benefits that you will get by referring to our Revision Notes on Class 12 CBSE Maths Chapter 6 Application of Derivatives:
The Class 12 CBSE Maths Chapter 6 notes on the "Application of Derivatives," available as a free PDF download, are an invaluable resource for both students and educators. These notes dive deep into the practical applications of calculus, revealing how derivatives are employed to analyse and optimise real-world scenarios. With clear explanations and comprehensive problem-solving, they empower students to grasp complex concepts related to rates of change, optimization, and tangent lines. These free PDF notes not only enhance mathematical proficiency but also demonstrate the significance of calculus in various fields like physics, economics, and engineering. In essence, these notes serve as a bridge between theory and practicality, preparing students for advanced studies and real-life problem-solving in mathematics.
