Applications of Derivatives

Derivatives are often used in mathematics. They’re useful for a variety of things, including determining the maxima and minima of a function, determining the slope of a curve, and even determining the inflection point. The derivative will be used in a few places, as shown below. The next sections go over each of these points in depth. The following are some of the most popular applications of derivatives:

• Calculating a Quantity’s Rate of Change
• Approximation Value Determination
• calculating the tangent and normal to a curve equation
• Identifying the Maxima and Minima, as well as the Point of Inflection
• Identifying Increasing and Decreasing Functions

Real Life Applications of Derivatives :

Automobiles: An odometer and a speedometer are always present in a vehicle. These two gauges work together to let the driver know how fast they’re going and how far they’ve traveled. Electronic versions of these gauges simply employ derivatives to convert data from the tyres to miles per hour (MPH) and distance sent to the electronic motherboard (KM).

Radar Guns: Every police officer who uses a radar gun is taking advantage of derivatives’ ease of use. When a radar gun is pointed at your car on the highway and fired, the gun can identify the time and distance at which the radar was able to hit a certain portion of your car. It is possible to compute the speed of the car from the radar gun by using derivatives.

Business : Derivatives have a wide range of uses in the business world. When data has been charted on a graph or data table, such as excel, one of the most essential applications is when the data has been plotted. After the data has been entered, it can be graphed, and derivatives can be used to determine the profit and loss point for various initiatives.

Applications of derivatives are widely used in various sciences :

• Physics 
• Biology 
• Economics 
• Chemistry 
• Mathematics
• Other subjects like Geology, Psychology etc.

Rate of Change of Quantity :

The rate of change is the amount of change in a quantity over time.

If a quantity y fluctuates with regard to another quantity x, satisfying y = f(x), the rate of change of y with respect to x is represented by dy/dx.

Lets understand with an example: If x = f(t) and y = g(t) be the two functions representing time then,

dy/dx=dydt/dxdt where dx/dt ≠ 0 which implies;

 dy/dx=dy/dt*dt/dx

The rate of change of y with respect to x at x =  x0 is represented by the value of dy/dx at x = x0 , i.e. (dy/dx)x= x0.

Approximation :

A function’s derivative can be used to obtain a function’s linear approximation at a particular value. Newton proposed the linear approximation approach, which entails first determining the value of the function at a specific point, then determining the equation of the tangent line to determine the function’s approximate value. The tangent function has the following equation:

L(x) = f(a) + f'(a)(x-a)

Maximum and Minimum Values :

Calculus has a lot of power when it comes to determining the maximum and minimum values of a function. Assume that f(x) is a function of x. The maximum, lowest, or inflection point of the function f refers to the value of x for which the derivative of f(x) with respect to x is equal to zero (x).

• If f(x) ≤ f(a) for every x in the domain when x = a, then f(x) has an Absolute Greatest value, and an is the place where f’s maximum value is found.
• If for every x in some open interval (p, q), f(x) ≤ f(a), then f(x) has a Relative Maximum value.
• When x=a and f(x) ≥ f(a) for every x in the domain, f(x) has an Absolute Minimal value, and an is the location where f’s minimum value is found.
• If for every x in some open interval (p, q), f(x) ≥ f(a), then f(x) has a Relative Minimum value.

Increasing and Decreasing :

A function’s derivative can be used to detect whether the function is increasing or decreasing at any point in its domain. The function is said to be rising on I if f′(x) > 0 at each point in the interval I. The function is said to be decreasing on I if f′(x) 0 at each point in the interval I. Because the derivative is zero or not present only at the function’s critical points, it must be positive or negative at all other locations where the function is present.

• If f'(x) > 0 for each x ∊ (p, q), f is increasing at [p, q]. 
• If f'(x) > 0, f is declining at [p, q]. 
• If f'(x)=0 for (p, q), f is constant

Tangent and Normal to the Curve :

A tangent to a curve is a line that touches it at one point and has the same slope as it.

A line perpendicular to a tangent to the curve is called a normal to the curve.

• The equation of a straight line which passes through the point (x1,y1) and has a slope m is given by : m=y-y1x-x1
• If two lines with the m1 slopes m2 and are at right angles then, 

m1.m2=-1

Conclusion :

In everyday life, derivatives are frequently employed to determine how much something is changing. The government employs them in population censuses, many disciplines, and even economics. Knowing how to employ derivatives, when to use them, and how to use them in real-life situations is an important aspect of any job./