Application of Derivatives

Understanding Application of Derivatives

Calculus is one of the essential topics in mathematics, which finds its usage in almost any subject which is somewhat related to mathematics. This includes physics and other branches of engineering. Moreover, other than the analytical application of derivatives, there is a ton of other real life application of differential calculus, without which many scientific proofs could not have been arrived at.

Some of the essential application of derivatives examples includes Maxima and Minima, normals and tangents to curves, rate of change of values, incremental and decremental functions, etc. Most of these are vital for future academics, as much as they are vital in this class.

Rate of Change of Quantities

The most important sub-topic of applications of partial derivatives is calculating the rate of change of quantities. In Physics, when we calculate velocity, we define velocity as the rate of change of speed with respect to time or ds/dt, where s = speed and t = time. Hence, rate of change of quantities is also a very essential application of derivatives in physics and application of derivatives in engineering.

Similarly, when a value y varies with x such that it satisfies y=f(x), then f’(x) = dy/dx is called the rate of change of y with respect to x. Also, f’(x0) = dy/dx x=x0 is the rate of change of y with respect to x=x0.

If two variables x and y vary w.r.t to another variable t such that x = f(t) and y = g(t), then via Chain Rule, we can define dy/dx as

\[\frac{dy}{dx}\] = \[\frac{dy}{dt}\] / \[\frac{dx}{dt}\], if \[\frac{dx}{dt}\] ≠ 0

Pop Quiz 1

1. Find Out the Rate of Change of Surface Area of a Cube When Length of Each Side of a Cube = 10cm and Rate of Change of Volume of Cube = 9 cc per second. 

    1. 3.6 (Answer)

    2. 5

    3. 10

    4. None of the above

Increasing and Decreasing Functions

Another usage of the application of derivatives formulas is increasing and decreasing functions. For example, let us take the below graph for analysing. 

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In the above graph, if we start from the origin and go towards positive infinity, we see that for each y, x is increasing. Hence, y = x2 is an increasing function for x>0. And if we arrive towards origin from negative infinity, we notice that for each two consecutive y values, their x values are decreasing. So, y = x2 is a decreasing function for x<0. 

There are certain rules due to which applications of derivatives solutions for increasing and decreasing functions become easier. Considering a function f is continuous and differentiable in [a,b], then f is

  1. Increasing in [a,b] if f’(x)>0 for all [a,b].

  2. Decreasing in [a,b] if f’(x)<0 for all [a,b].

  3. Constant in [a,b] if f’(x)=0 for all [a,b].

Activity

Take a notebook and try to prove f(x) = 9x – 5 is increasing on all real values to understand more about application of partial differentiation.

Maxima and Minima

Another example of derivatives in real life is the calculation of maxima and minima. It is also one of the widely used applications of differentiation in physics.

Maxima and minima are useful in finding the peak points in graphs where a graph exhibits its maximum or its minimum value locally within a given region. Some rules to find these values to help you to find application of derivatives NCERT solutions are:

  1. If x = b, b is called the Absolute Maximum if for a graph, f(x) <= f(b) for the whole domain. 

  2. If x = b, b is called the Local Maximum if for a graph, f(x) <= f(b) for a particular domain, say [m,n].

  3. If x = b, b is called the Absolute Minimum if for a graph, f(x) >= f(b) for the whole domain. 

  4.  If x = b, b is called the Local Minimum if for a graph, f(x) >= f(b) for a particular domain, say [m,n].

Pop Quiz 2

1. What are the Values of x at Maxima and Minima for y = x2?

    1. Maxima at 0, Minima at 0

    2. Maxima at positive infinite, Minima at negative infinite

    3. No maxima, Minima at 0 (Answer)

    4. None of the above

So, this was all about applications of derivatives and their real life examples. We hope that our concise guide will help you in finding all NCERT solution of application of derivatives. For more such tutorials and guides on other topics, visit the Vedantu website today or download our app.

FAQ (Frequently Asked Questions)

1. What are Some of Applications of Derivatives in Real Life Examples?

Ans. Almost all the applications have some real life usage when it comes to partial derivatives and absolute derivatives. When a value y varies with x such that it satisfies y=f(x), then f’(x) = dy/dx is called the rate of change of y with respect to x. Also, f’(x0) = dy/dx x=x0 is the rate of change of y with respect to x=x0. Rate of change of values is a significant application of differentiation example, which is used broadly in physics and other engineering subjects. 

2. What are Increasing and Decreasing Functions?

Ans. Another important NCERT application of derivatives solutions is the concept of increasing and decreasing functions. The rules with which we can determine if a function is one of the above are:

Considering a function f is continuous and differentiable in [a,b], then f is

  1. Increasing in [a,b] if f’(x)>0 for all [a,b].

  2. Decreasing in [a,b] if f’(x)<0 for all [a,b].

  3. Constant in [a,b] if f’(x)=0 for all [a,b].

For example, y = x2 is an increasing function for x>0 and a decreasing function for x<0. 

3. What are Maxima and Minima?

Ans. Another one of examples of derivatives in real life is the concept of maxima and minima. The rules to find such points on a graph are: 

  1. If x = b, b is called the Absolute Maximum if for a graph, f(x) <= f(b) for the whole domain. 

  2. If x = b, b is called the Local Maximum if for a graph, f(x) <= f(b) for a particular domain, say [m,n].

  3. If x = b, b is called the Absolute Minimum if for a graph, f(x) >= f(b) for the whole domain.

4. What is the Application of Derivatives of Trigonometric Functions? Describe with One Example.

Ans. Tangents and normals are very important applications of derivatives. Inside a graph, if we draw a line that just touches the curve and does not intersect it, that line is called a tangent. Similarly, a normal is a line which is perpendicular to a tangent. In the figure below, the curve is the green line, and the other two lines are marked.  

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The formula of a tangent is given by y – y1 = f’(x1)(x-x1), while the formula for a normal is (y – y1) f’(x1) + (x-x1) = 0.