Applications Of Derivatives

Applications Of Derivatives: In this article, we are going to learn about the derivative applications in real life and in other sections. Limits and derivatives are high-level concepts which are taught in higher classes and requires a detailed understanding of it. 

You must have learned to find the derivative of different functions, like, trigonometric functions, implicit functions, logarithm functions, etc. Now we need to learn its applications with respect to mathematical concepts and in real life scenarios.

Definition Of Derivative

The derivative is defined as the rate of change of one quantity with respect to another. In terms of functions, the rate of change of function is defined as dy/dx = f(x) = y’. The ratio of dy/dx is used as one of the applications of derivatives in real life and in various aspects. This is the general representation or formula of the derivative

Applications

The concept of derivatives has been used in small scale and large scale, such as in engineering, physics, etc. Whether its the matter of change of temperature or rate of change of shapes and sizes of an object depending upon the conditions, etc, the concept of derivatives have been used in many ways. Now, let us see the applications based on derivative concepts;

Rate of Change of a Quantity

This is the general and most important application of derivative. 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 cube.

Increasing and Decreasing Functions

To find that a given function is increasing or decreasing or constant, say in a graph, we use derivatives. If f is a function which is continuous in [p,q] and differential on the open interval [p,q], then,

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 constant function in [p,q], if f′(x)=0 for each x ∈ (p,q)

Tangent and Normal To a Curve

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

Let the tangent meet the curve at (x1,y1). Now from the straight line equation which passes through a point having slope m, could be written as;

y – y1 = m(x – x1)

We can see from the above equation, the slope of the tangent to the curve is y=f(x) and at the points (x1,y1), it is given as dy/dx at (x1,y1) = f’(x).

Therefore, the equation of the tangent to the curve at (x1,y1) can be written by;

y – y1 = f’(x1)(x-x1)

And the equation of normal to the curve is given by;

y – y1 = \(\frac{-1}{f'(x_{1})}\) (x-x1)

Or (y – y1) f’(x1) + (x-x1) = 0

Maxima and Minima

To calculate the highest and lowest point of the curve in a graph or to know its turning point, derivative function is used.

When x= a, if f(x) ≤ f(a) for every x in the domain, then f(x) has an Absolute Maximum value and the point a is the point of the maximum value of f.

When x = a, if f(x) ≤ f(a) for every x in some 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 then f(x) has an Absolute Minimum value and the point a is the point of the minimum value of f.

When x = a, if f(x) ≥ f(a) for every x in some open interval (p, q) then f(x) has a Relative Minimum value.

Applications of Derivatives

Approximation or To Find Approximate Value

To find a very small change or variation of a quantity, we can use derivatives to give the approximate value of it. The approximate value is represented by delta △.

Suppose change in the value of x, dx = x then,

dy/dx = △x = x.

Since the change in x, dx x therefore, dy ≈ y.

Also Read:

Application of Derivatives in Real Life

  • To calculate the profit and loss in business using graphs.
  • To check the temperature variation
  • To determine the speed or distance covered such as miles per hour, kilometre per hour etc.
  • Derivatives are used to derive many equations in Physics.
  • In the study of Seismology like to find the range of magnitudes of the earthquake.

By solving the application of derivatives problems, your concepts for these applications will be understood. Let us give an example.

Examples

Example: Show that the function f(x) = x3 – 2x2 + 2x, x ∈ Q is increasing on Q.

Solution: f(x) = x3 – 2x2 + 2x

By differentiating both the sides, we get,

f’(x) = x2 – 2x + 2 > 0 for every value of x

Therefore, f is increasing on Q.

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Practise This Question

(96 ÷ 12) + 14 x (12 + 8) ÷ 2