You must have studied the concept of derivatives in your previous classes, such as derivatives of trigonometric functions, implicit functions, logarithmic functions, etc. In this section, you will learn how derivatives can be applied to mathematical concepts and scenarios in real life. Derivatives have a wide range of applications, not only in mathematics and real life, but also in fields such as science, engineering, and physics. This is an important topic covered in Class 12 Maths.

Table of Contents:

[Application of Derivatives](#Application of Derivatives)

[Rate of Change](#Rate of Change)

‘Increasing and Decreasing Function’

[Tangent and Normal](#Tangent and Normal)

Minima and Maxima

Monotonicity

Approximation

‘+ Point of Inflection’

[Daily Life Application](#Daily Life Application)

[Video Lessons](#Video Lessons)

[Solved Examples](#Solved Examples)

[Practice Questions](#Practice Questions)

FAQs

Applications of Derivatives in Maths

The derivative is defined as dy/dx = f(x) = y', which is the rate of change of one quantity with respect to another in terms of functions.

The concept of derivatives has been employed both on a small and large scale. It can be used in a variety of ways, such as to measure changes in temperature or the rate of change of the shape and size of an object depending on certain conditions.

Rate of Change of a Quantity

The general and most important application of derivatives is to check the rate of change of a certain variable with respect to another. For example, to check the rate of change of the volume of a cube with respect to its decreasing sides, we can use the derivative form as dy/dx, where dy represents the rate of change of volume of cube and dx represents the change of sides of the cube.

Increasing and Decreasing Functions

If the derivative of f is positive in the open interval (p, q), then the function f is increasing in [p, q]. If the derivative of f is negative in the open interval (p, q), then the function f is decreasing in [p, q]. If the derivative of f is zero in the open interval (p, q), then the function f is constant in [p, q].

‘f is increasing at [p, q] if f’(x) > 0 for each x ∈ (p, q)’

‘f is decreasing at [p, q] if f’(x) < 0 for each x ∈ (p, q)’

f is decreasing at [p, q] if f’(x) < 0 for all x ∈ (p, q)

‘f is a constant function in the interval [p, q] if its derivative, f’(x), is equal to 0 for all x in (p, q).’

Tangent and Normal to a Curve

A tangent is a line that touches the curve at a point and doesn’t cross it, whereas a normal is perpendicular to that tangent.

Let the tangent meet the curve at P(x1, y1).

Now the straight-line equation which passes through a point with slope m can be written as:

y - y1 = m(x - x1)

Therefore, the slope of the tangent line to the curve y = f(x) at the point P(x1, y1) can be calculated by taking the derivative of the function at x1, that is, f’(x1).

The equation of the tangent line at P(x1, y1) is: $$y-y_1 = \frac{dy}{dx}\bigg|_{(x_1,y_1)}(x-x_1)$$

y - y1 = f’(x - x1)

The equation of the normal to the curve is:

y - y1 = [-1/f’(x1)] * (x - x1)

Either

(x - x1)f’(x1) + (y - y1) = 0

Maxima and Minima

To determine the highest and lowest point of a curve, or to find its turning point, the derivative function is utilized.

If f(x) is less than or equal to f(a) for every x in the domain when x = a, then f(x) has an Absolute Maximum value and the point a is the point of the maximum value of f.

If f(x) is less than or equal to f(a) for every x in the open interval (p, q), then f(x) has a Relative Maximum value when x = a.

If f(x) ≥ f(a) for every x in the domain and x=a, then f(x) has an Absolute Minimum value and the point a is the point of the minimum value of f.

If f(x) is greater than or equal to f(a) for every x in the open interval (p, q), then f(x) has a Relative Minimum value when x = a.

Monotonicity

f(x) = ex, f(x) = nx, and f(x) = 2x + 3 are all examples of monotonic functions, which are functions that are either increasing or decreasing in their entire domain.

Functions which are neither increasing nor decreasing in their domain are said to be non-monotonic

For example:

• f(x) = sin x
• f(x) = x²

Monotonicity of a Function at a Point

A function is said to be monotonically decreasing at x = a if f(x) is decreasing as x approaches a.

f(x + h) > f(a) for a small positive h

‘f’(x) will be positive if and only if the function is increasing.

‘f’(x) will be negative if and only if the function is decreasing.

‘f’(x) will be equal to zero when the function is at its local maxima or minima’

Finding Approximate Value

The approximate value of a very small change or variation of a quantity can be found using derivatives, which is represented by the symbol $\Delta$.

If x changes, then dx = x.

dy/dx = \frac{d}{dx}(x) = 1

Since the change in x, $\frac{dy}{dx} \approx \frac{y}{x}$ therefore, $dy \approx y$.

Point of Inflection

If $f’(x_0) = 0$ or $f’’(x_0)$ does not exist at points where $f’(x_0)$ exists and if $f’’(x)$ changes sign when passing through $x = x_0$, then $x_0$ is called the point of inflection.

If f”(x) < 0, x ∈ (a, b), then the curve y = f(x) is concave downward.

If f"(x) > 0, for x ∈ (a,b), then the curve y = f(x) is concave upwards in (a,b).

For example: f(x) = sin(x)

f(x) = cos(x)

f(x) = sin(x) = 0, x = nπ, n ∈ \mathbb{Z}

Application of Derivatives in Real Life

To graphically represent the profit and loss in business.

To check the temperature fluctuation.

To calculate the speed or distance travelled, such as miles per hour, kilometres per hour, etc.

Derivatives are often used to calculate many equations in Physics.

In the study of Seismology, researchers are interested in finding the range of magnitudes of earthquakes.

By understanding the concepts behind the applications of derivatives, one can solve the problems related to derivatives in a better manner.

Video Lessons

Uses of Derivatives

![Greatest Integer Function]()

Important Topics

Important Questions to Check

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Examples of Applications of Derivatives

Example 1:

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This sentence has been rewritten.

The derivative of the function $f(x) = x^3 - 2x^2 + 2x$, $x \in \mathbb{Q}$, is $f’(x) = 3x^2 - 4x + 2$. Since $f’(x) > 0$ for all $x \in \mathbb{Q}$, the function is increasing on $\mathbb{Q}$.

Given:

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Solution:

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f(x) = x^3 - 2x^2 + 2x

By differentiating both sides, we obtain:

f(x) = 3x^2 - 4x + 2 > 0 for every value of x

Therefore, f is increasing on Q.

Example 2:

Find the coordinates of the point where the tangent to the curve y=x2− 5x + 5, which is parallel to the line 2y = 4x + 1, passes through.

Given:

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Solution:

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dy/dx = 2x - 5, x = x1 = 2

dy/dx|x=x1 = 2(2) - 5 = 4 - 5 = -1

2x1 ≠ 7

x1 = 3.5

y1 = $\frac{49-70+20}{4}$ = -$\frac{1}{4}$

y + ¼ = 2x - 7

4y + 1 = 8x - 28

4y = 8x - 29

8x - 4y = 29

x = ⅛ and y = -7 satisfies the equation x + y = -⅝

Example 3:

The tangent to the curve, (y=x{{e}^{{{x}^{2}}}}) passing through the point ((1, e)) also passes through another point. Find it.

Given:

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Solution:

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$$\frac{dy}{dx}=e^{x^2}+x\cdot e^{x^2} \cdot 2x$$

At x = 1, the slope of the tangent is m = 3e

Equation of Tangent: \frac{dy}{dx} = \frac{f(x) - f(a)}{x - a}

y - e = 3e(x - 1)

y = 3e^x - 2e

The point (4/3, 2e) lies on it.

Practice Problems

1. Use the Differential Approximation Method to find the approximate value of $\sqrt{26}$.

Find the local maxima or minima, if any, for $f(x) = \frac{1}{x^2 + 2}$

3. Proof that f(x) = e–x is a strictly decreasing function on R:

Let x<y, then,

f(x) = e–x

f(y) = e–y

e–x > e–y

Therefore, f(x) > f(y), and f(x) is strictly decreasing on R.

Frequently Asked Questions on Applications of Derivatives

What are the Uses of Derivatives?

The applications of derivatives are:

• Optimization
• Equilibrium
• Modeling of Systems
• Risk Management
• Financial Analysis

To calculate the rate of change of one quantity with respect to another changing quantity.

To determine the maximum, minimum, and saddle points of a function.

To identify whether a function is concave or convex.

For Approximations

To find the tangent and normal of a function at a point, use derivatives to calculate the slope of the tangent line at that point.

Let y = f(x) be a function for which we have to find a tangent at a point (x1, y1), then we find dy/dx at (x1, y1). To determine whether a function is increasing or decreasing using derivatives, if dy/dx > 0, then the function is increasing, and if dy/dx < 0, then the function is decreasing.

If $f$ is a continuous function in the closed interval $[p, q]$ and a differentiable function in the open interval $(p, q)$, then

f is decreasing at [p, q] if f'(x) < 0 for each x ∈ (p, q)

f is decreasing at [p, q] if f'(x) < 0 for each x ∈ (p, q)

f is increasing at [p, q] if f'(x) > 0 for each x ∈ (p, q)

If you want to find the point of inflection using derivatives, you need to find the value of $x$ for which $f’(x)=0$ for each $x$ in the interval $(p, q)$, where $f$ is a constant function in the interval $[p, q]$.

If f(x) is a differentiable function, then it is said to be continuous.

“Concave up at a point $x=a$ if and only if $f(x) > 0$ at $a$”

“Concave down at a point x = a, if and only if f(x) < 0 at a”

Here, f''(x) is the second order derivative of the function f(x).