The derivative can be defined as the rate of change of one variable with reference to another. The derivative, in terms of functions, can be expressed as dy/dx = f(x) = y’. Various Applications of derivatives in the field of mathematics are:

- To find increasing and decreasing functions
- To find equations of tangent and normal to a curve at a point
- To determine the rate of change of quantities
- To find turning points in a graph that helps in determining the maximum and minimum values
- To find approximate values of certain quantities

## Increasing and Decreasing Functions

We can identify if a function is increasing or decreasing with the help of derivatives. Now, let f be a continuous function on [a,b] and differentiable on the open interval (a,b), Then

- f is increasing in [a,b] if f’(x)>0 for each x ∈ (a,b)
- f is decreasing in [a,b] if f’(x)<0 for each x ∈ (a,b)

f is a constant function in [a,b] if f’(x) = 0 for each x ∈ (a,b)

## Tangents and Normals

We use differentiation to find the equation of the tangent line and the normal line to a curve at a given point. A tangent line to a plane curve at any given point is a straight line just touching the curve at the given point. The slope of the tangent line is equal to the derivative of the curve at the same given point. However, a line that intersects the curve at a certain point is called a normal at that point. A normal is perpendicular to the tangent line at that given point.

For a straight line having finite slope m and passing through a given point (x0,y0), the equation is written as

y-y0 = m (x-x0)

At the point (x0,y0), keep in mind, to the curve y = f(x), the slope of the tangent is written as dy/dx at x0,y0 (=f’(x0)). Therefore, to the curve y=f(x), the equation of the tangent at (x0,y0) is written as

y-y0 = f’(x0)(x-x0)

Also, since the normal is perpendicular to the tangent, the slope of the normal to the curve y = f(x) at (x0,y0) is -1/f’(x0), if f’(x0) ≠ 0. Therefore, the equation of the normal to the curve y = f(x) at (x0,y0) is given by

y-y0 = -1/f’(x0) (x-x0)

i.e. (y-y0)f’(x0)+(x-x0) = 0

If the tangent line to the curve y = f(x) makes an angle 𝛉 with x-axis in the positive direction, then dy/dx= slope of tangent = tan 𝛉.

## Rate of Change Of Quantities

Whenever one quantity y varies with another quantity x, satisfying y = f (x), then dy/dx represents the rate of change of y with respect to x at x = x0. Now let’s consider that the two given variables X and Y are varying with respect to some other variable let’s say T, i.e., if x = f(t) and y = g(t), then by Chain Rule dy/dx = dy/dt/dx/dt , if dx/dt ≠ 0. This implies the rate of change of y with respect to x can be formulated using the rate of change of y and that of x both with respect to t. dy/dx is positive if the increase of y is directly proportional to x and is negative if the increase of x is inversely proportional to y.

## Turning Points in the Graph

The concept of derivatives is used to calculate the maximum or minimum values of various functions. We shall, in fact, find the turning points of the graph of a function and thus find points at which the graph reaches its highest or lowest locally. Now let f be a continuous and differentiable function on a closed interval I = [a,b] and let c be any interior point of I. Then

- f’(c) = 0 if f attains its absolute maximum value at c.
- f’(c) = 0 if it attains its absolute minimum value at c.
- When the sign of f’(x) or the sign of slope does not change as the graph proceeds with c, then point c is called the Point of Inflection.

## Approximation Values

One of the Applications of derivatives is to approximate values of various quantities. Let us take a function f such that y = f(x). Let ∆x be considered a small increase in x. Therefore the increment in y corresponding to the increase in x, indicated as ∆y, is given by ∆y = f(x+∆x ). Thus we can say that.

- dx = ∆x is used to define the differential of x, denoted as dx.
- dy = f’(x) dx or dy =(dy/dx)∆x is used to define the differential of y, denoted as dy.

Now if dx = ∆x, when compared to x is relatively small, dy then is a good approximation of ∆y and can be implied as dy = ∆y. The dependent variable’s differential is not equal to the variable’s increment, but the independent variable’s differential is equal to the variable’s increment.

## Applications of Derivatives In Real Life

There are multiple events in real life that can be studied by applying the concept of derivatives. Some of them, namely:

- Fluctuation of the stock market over time can be used to predict future stocks.
- The growth rate of a population over time
- Changes in temperature as a function of location
- Determining the speed or distance covered

## Conclusion

We studied the various applications of derivatives. It is an extremely important concept and is useful in various aspects of life. A derivative can be defined as an expression that gives the rate of change of a function with respect to an independent variable. In simple terms, derivatives represent the rate of change, and in mathematics, they can be applied to various circumstances. For example, velocity is said to be the rate of change of distance over time and acceleration is defined as the rate of change of velocity over time. Therefore derivative functions can be used to calculate the velocity or acceleration of a given object at any place in time.