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Applications of Derivatives
Applications of derivatives are varied not only in maths but also in real life. To give an example, derivatives have various important applications in Mathematics such as to find the Rate of Change of a Quantity, to find the Approximation Value, to find the equation of Tangent and Normal to a Curve, and to find the Minimum and Maximum Values of algebraic expressions.
Derivatives are vastly used across fields like science, engineering, physics, etc. In this article, we will learn the application of derivatives in real life.Let us learn about these applications of derivatives in detail.
Applications of Derivatives in Maths
In maths, derivatives have wide usage. They are used in many situations like finding maxima or minima of a function, finding the slope of the curve, and even inflection point. A few places where we will use the derivative are given below. And each of it is explained in detail in the following sections. The most common usage of application of derivatives is seen in:
 Finding Rate of Change of a Quantity
 Finding the Approximation Value
 Finding the equation of a Tangent and Normal To a Curve
 Finding Maxima and Minima, and Point of Inflection
 Determining Increasing and Decreasing Functions
Derivative for Rate of Change of a Quantity
Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is
(f(a + h)f(a))/h
Now using the definition of derivative, we can write
\(f′(a)=\lim_{h→0}\frac{f(a+h)−f(a)}{h}\)
which is also the instantaneous rate of change of the function f(x) at a.
Now, for a very small value of h, we can write
f'(a) ≈ (f(a+h) − f(a))/h
or
f(a+h) ≈ f(a) + f'(a)h
This means, if we want to find the small change in a function, we just have to find the derivative of the function at the given point, and using the given equation we can calculate the change. Hence the derivative gives the instantaneous rate of change of a function within the given limits and can be used to find the estimated change in the function f(x) for the small change in the other variable(x).
Approximation Value
Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value to the function. The equation of the function of the tangent is
L(x) = f(a) + f'(a)(x−a)
The tangent will be a very good approximation to the function's graph and will give the closest value of the function. Let us understand this with an example, we can estimate the value of √9.1 using the linear approximation. Here we have the function: f(x) = y = √x. We will find the value of √9 and using linear approximation, we will find the value of √9.1.
We have f(x) = √x, then f'(x) = 1/(2√x)
Putting a = 9 in L(x) = f(a) + f'(a)(x−a), we get,
L(x) = f(9) + f'(9)(9.1−9)
L(x) = 3 + (1/6)0.1
L(x) ≈ 3.0167.
This value is very close to the actual value of √(9.1)
Hence by using derivatives, we can find the linear approximation of function to get the value near to the function.
Tangent and Normal To a Curve
The equation of tangent and normal line to a curve of a function can be calculated by using the derivatives. If we have a curve of a function and we want to find the equation of the tangent to a curve at a given point, then by using the derivative, we can find the slope and equation of the tangent line. A tangent is a line to a curve that will only touch the curve at a single point and its slope is equal to the derivative of the curve at that point. The slope(m) of the tangent to a curve of a function y = f(x) at a point \((x_1, y_1)\) is obtained by taking the derivative of the function (m = f'(x) ).
By finding the slope of the tangent line to the curve and using the equation \(m = (y  y_1)/(x  x_1) \), we can find the equation of the tangent line to the curve. Similarly, we can find the equation of the normal line to the curve of a function at a point. This normal line will be normal(perpendicular) to the tangent line. Hence the slope of the normal line to a curve of a function y = f(x) at a point \((x_1, y_1)\) is given as follows.
n = 1/m =  1/ f'(x)
And by using the equation \(1/m = (y  y_1)/(x  x_1) \) we can find the equation of the normal line to the curve.
Maxima, Minima, and Point of Inflection
Application of derivatives is also helpful in finding the maxima, minima, and point of inflection of a curve. Maxima and minima are the peaks and valleys of a curve, whereas the point of inflection is the part of the curve where the curve changes its nature(from convex to concave or vice versa). We can find the maxima, minima, and point of inflection by using the firstorder derivative test. According to this test, we first find the derivative of the function at a given point and equate it to 0, i.e., f'(c) = 0, (here we have found the slope of the curve equal to 0, which means it is a line parallel to the xaxis). Now if the function is defined in the given interval, then we check the value of f'(x) at the points lying to the left of the curve and to the right of the curve and check the nature of the f'(x), then we can say, that the given point is maxima or minima based on the below conditions.
 Maxima when the slope or f’(x) changes its sign from +ve to ve as we move via point c. And f(c) is the maximum value.
 Minima when the slope or f’(x) changes its sign from ve to +ve as we move via point c. And f(c) is the minimum value.
 Point C is called the Point of inflection when the sign of slope or sign of the f’(x) doesn’t change as we move via c.
Increasing and Decreasing Functions
By using derivatives, we can find out if a function is an increasing or decreasing function. The increasing function is a function that seems to reach the top of the xy plane whereas the decreasing function seems like reaching the downside corner of the xy plane. Let us say we have a function f(x) which is differentiable within the limits (a, b). Then we check any two points on the curve of the function.
 If at any two points \(x_1\) and \(x_2\) such that \(x_1\) < \(x_2\), there exists a relation \(f(x_1)\) ≤ \(f(x_2)\), then the given function is increasing function in the given interval, and if \(f(x_1)\) < \(f(x_2)\), then the given function is strictly increasing function in the given interval.
 And, if at any two points \(x_1\) and \(x_2\) such that \(x_1\) < \(x_2\), there exists a relation \(f(x_1)\) ≥ \(f(x_2)\), then the given function is decreasing function in the given interval and if \(f(x_1)\) > \(f(x_2)\), then the given function is strictly decreasing function in the given interval
Related Topics on Applications of Derivatives:
Important Notes on Applications of Derivatives:
 The application of derivatives is used to find the rate of changes of a quantity with respect to the other quantity.
 The equation of tangent and normal line to a curve of a function can be calculated by using the derivatives.
 Derivative of a function can be used to find the linear approximation of a function at a given value.
Solved Examples

Example 1: Find the maxima and minima of a function:y = 2x^{3}  3x^{2} + 6 using the formula for applications of derivatives.
Solution
Given function: y = 2x^{3}  3x^{2} + 6
Using the second order derivative test we can find the maxima and minima of a function:
Taking first order derivative:
y = 2x^{3}  3x^{2} + 6 (eq 1)
Differentiate both of sides (eq 1), w.r.t  x.
⇒ dy/dx = d/dx (2x^{3})  d/dx (3x^{2}) + d/dx (6)
⇒ dy/dx = 6x^{2}  6x + 0
⇒ dy/dx = 6x^{2}  6x (eq 2)
Putting dy/dx = 0 to find critical points.
⇒ 6x^{2}  6x = 0
⇒ 6x (x  1) = 0
⇒ x = 0,1
The critical points are 0 & 1.
Differentiate both sides of (eq 2), w.r.t  x.
⇒ d^{2}y/dx^{2} = d/dx (6x^{2})  d/dx (6x)
⇒ d^{2}y/dx^{2} = 12x  6
Now, put the value of x and find the max or min value.
At x = 0, d^{2}y/dx^{2} = 12(0)  6 = 6, hence x = 0 is a maxima
At x = 1, d^{2}y/dx^{2} = 12(1)  6 = 6, hence x = 1 is a minima
Answer: The maxima of the function is at x = 0 and minima of the function is at x = 1.

Example 2: If f(4) = 5, and f'(4) = 9, find the value of f(4.1) using the application of derivatives.
Solution:
Given: f(4) = 5, and f'(4) = 9
To find: f(4.1)
Using formula for approximation:
L(x) = f(a) + f'(a)(x−a)
L(x) = 5 + 9 (4.14)
L(x) = 5 + 9(0.1)
L(x) = 5.9
Answer: The value of f(4.1) is 5.9.
FAQs on Applications of Derivatives
What Is the Application of Derivatives in Math?
In maths, derivatives have wide usage. They are used in many situations like finding maxima or minima of a function, finding the slope of the curve, and even inflection point. A few places where we will use the derivative are given below and then explained one by one in the following sections. The most common usage of the application of derivatives is seen in the following areas.
 Finding Rate of Change of a Quantity
 Finding the Approximation Value
 Finding Tangent and Normal To a Curve
 Finding Maxima and Minima, and Point of Inflection
 Determining Increasing and Decreasing Functions
What Is the Application of Differentiation in Real Life?
Differentiation has wide usage in real life. A few of the applications are:
 In business, differentiation is used to find profit and loss for the future of the investment using graphs.
 Temperature variations are also calculated by using differentiation.
 It is used to calculate the rate of change of distance of a moving body with respect to time.
What Are the Applications of Differential Calculus?
Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. The equation of tangent and normal line to a curve of a function can be calculated by using the derivatives. Derivative of a function can be used to find the linear approximation of a function at a given value. Derivatives are also helpful in finding the maxima, minima, and point of inflection of a curve.
Which Topics Come Under Application of Derivatives?
The following chapters come under the application of derivative:
 Rate of Change of Quantities.
 Approximations.
 Increasing and Decreasing Functions.
 Maxima and Minima.
 Tangents and Normals.
 Increasing and Decreasing Function.
Why Are Derivatives in Math Important?
Derivatives represent a rate of change. In mathematics, a rate of change can be applied to many circumstances. For instance, acceleration is the rate of change in velocity. Therefore, a derivative function can be used to determine the acceleration of an object when the velocity over time is given.
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