Maths Applications of Derivatives: Overview, Questions, Preparation

The concept of derivatives has broad uses in some of the major fields such as science, engineering, economics, mathematics, biomedical sectors, optics, astronomy and many more. Specifically talking about mathematics, it has proved to be the backbone of major problems related to linear and differential equations. Let us dive into the detailed aspects of its core uses:

1. Physics:

Derivatives can be used to determine the rate of change of quantities whichare used to solve numerical problems. This change is represented by the equation dy/dx i.e. rate of change of y with respect to x.

For example:

V = ds/dt

Where,

S = displacement

T = time

V = velocity

Similarly, acceleration:

a = dv/dt = d^2.s/d.t^2

2. Slope of a Curve:

The concept of slope is used to find the steepness of the curve as well as the exact location of points on the curve lying over the graph. The equation can be defined as:

M = dy/dx

Where,

M = slope

Dy/dx = the rate of change of y with respect to x.

3. Curve Sketching

Derivatives can also be used to find the shape and direction of the graph. For this, the conditions are as follows:

• If f(x)>0: curve is pointing in the upward direction.
• If f(x)< 0: curve is pointing in the downward direction.

4. Maxima and Minima

Maxima and Minima is considered to be one of the most important topics under this chapter. Derivatives are used to calculate the local maxima and minima of a particular function. The conditions specified for this are:

If f(x)>0, point is a local minimum.

If f(x)< 0, point is a local maximum.

If f(x)= 0, it is a critical point.

5. Economics, Finance and Business:

Business strategies require the use of derivatives to find shortcomings in their model and on the basis of that, calculate:

• Profit and Loss
• Marginal Cost (MC = dc/dq)
• Marginal Revenue (MC = dr/dq)
• Supply Chain
• Demand in the market
• Market trends

 

6. Machine Learning and Data Science:

This concept has major uses in the field of computer science too and is used in the fields of principal component analysis (PCA), gradient descending, big data analytics (BDA), computer networking, compiler design and training models, data algorithms, discrete mathematics, artificial intelligence, data mining, coding and programming, etc.

7. Health Sector:

Derivatives also have major uses in the biomedical industry. Doctors calculate the exact amount of overdose and concentration using the derivative formula:

Absorption rate – Elimination rate

This helps them provide optimal dosage to the patients without causing any harm to their bodies which can lead to speedy recoveries.