## Introduction

Derivatives is the most important topic of mathematics. It is often involved in numerous types of calculations. It helps determine the rate of change of function with respect to a variable. The application of derivatives is helpful to determine the rate of change of a quantity, calculate the approximate value, find the equation of a tangent and normal to a curve, finding maxima and minima derivatives. They play a huge role in the fields of physics and engineering but they are also applicable in real life.

## What are Derivatives?

Derivative is the rate of change of a function with respect to a variable. Numerically, it can be

y’ = dy/dx where y is the function whose value changes with the change in x.

### Relevance of Application of Derivatives in Mathematics

Application of the derivatives is highly relevant in the numerous topics of mathematics. As mentioned above, it is highly applicable for multiple mathematical algebraic applications. A few main areas where the application of derivatives is used are as follows:

- For determining the equation of tangent and normal to a curve
- Determining the maxima and minima along with the point of inflection
- Determining the increasing and decreasing functions

### For determining the equation of tangent and normal to a curve

The tangent is a line that touches a curve at one point such that it is perpendicular to the radius and the perpendicular to that tangent is known as the normal.

The derivatives can be used to calculate the equation of tangent and normal to a curve. If you have come across a situation in which the curve function is there and have to find the derivative to the curve, you can appropriately use the derivatives to get a solution. Through the derivatives, we can determine the slope and the equation of the tangent line and can get our answer. We can simply find the equation to the tangent line by taking the tangent of the function (m = F(x)) which yields the slope(m) of the tangent to a curve of a function y = f(x) at a point [x1, y1 ]

We may obtain the equation of the tangent line to the curve by determining the slope of the tangent line to the angle and applying the equation m = [(y-y1)/(x-x1)]. Similarly, we may obtain the equation of the normal line to a function’s curve at a given location. This normal line will be perpendicular (normal) to the tangent line.

### Determining the maxima and minima along with the point of inflexion

The peaks and troughs of a curve are known as maxima and minima, respectively, whereas the inflexion is that point of the curve where the angle changes its character (from convex to concave or vice versa). Derivatives may also be used to locate the maxima, minima, and inflection points. So, the derivative application is also accessible to determine the maxima or minima. We can evaluate the maxima, minima, and point of inflection using the first-order derivative test.

For this, we first find the function’s derivative at a particular point and equate it to zero, i.e., if f’(c) = 0. (here, we have seen the slope of the curve equal to 0). Therefore, assuming the function is defined in the given interval, we can examine the value of f (x) at the points lying to the left and right of the curve and the nature of the f(x). Based on the requirements listed below, we can declare whether the given point is maximum or minimum.

- Maxima occurs when the slope, or f’ (x), changes signs from positive to negative as we go through point c. And the maximum value is f(c).
- Point C is the Point of Inflection when the slope sign or f’(x) does not change as we walk along with c.
- Minima occurs when the slope, or f’ (x), changes the sign from -ve to +ve as we go through point c. And f(c) is the smallest value.

### Determining the Increasing and decreasing functions.

We may determine if a function is rising or decreasing by utilizing derivatives. The increasing function seems to reach the top of the x-y plane, whereas the decreasing function comes from the bottom corner of the x-y plane. Assume we have a function f(x) differentiable within certain bounds (a, b). Then we examine any two points on the function’s curve.

Suppose there exists a connection f(x1)≤f(x2) between any two points x1 and x2. In that case, the provided function is an increasing function in the given interval, and if f(x1)<f(x2), then the supplied function is a strictly increasing function in the specified interval.

And, if there exists a connection f(x1)≥ f(x2) between any two points x1 and x2, then the provided function is a decreasing function in the given interval.

### Conclusion

Derivatives are the most important topic of mathematics. They have been massively crucial for engineering activities. Although, we can see several exercises where we may find the derivatives’ application. You can apply the application of the derivative and quickly get your answer in some specific algebraic situations. Whether you are stuck over a tangent and normal to a curve or maxima and minima, the applications of derivative is the solution for your problem. The application of the derivatives is accessible for numerous mathematical issues, but the above-suggested situation is most specific.