# Derivatives

Derivatives are the varying rate of change in function with respect to an independent variable.

## What is the meaning of derivatives?

The meaning of derivatives, in business mathematics, is the rate of change of a function concerning an independent variable that is used in varying quantity. The rate of change of derivatives is not constant and it is used to measure the sensitivity of the dependent variable with respect to the independent variable.

In simple words, the meaning of derivatives is the instantaneous change of quantity with respect to another. For example, a car takes ‘t’ minutes to go from point a to point b. We can know how long a car takes to move from point a to point c by its velocity. The velocity of a car is as follows:

Velocity = d(x) / d(t), where x is the total distance and t is the time taken by the car to move from one point to another.

## Calculus

The process of finding the derivate is called differentiation and its inverse is known as anti-differentiation. The derivative of a function y=f(x) is f’(x), which is the measure at which the value of y is dependent on the variable x. When an infinitesimal change to x is dx, then the derivative of y will be displayed as dy / dx. You can read it as “dy over dx”.

For example, let us assume that y is a dependent variable and x is an independent variable. A change in the value of x is dx, and any change in x will bring a change in y. Let us also assume f(x) to be the function whose value depends on x. You can find the derivative formulas by following these steps.

• Bring a small change to the value of x so that the function will become f(x+h). Here, h is the small change added to x.
• The change in the value of the function will be denoted as f(x+h) – f(x).
• The rate of change in f(x) will be denoted as dy / dx = h0f(x+h) – f(x) / h

Here, d(x) is ignored as the value is negligible.

## Derivative Formulas

There are many derivative formulas that have their use in linear functions, exponential functions, and logarithmic functions. Some of the derivative formulas are listed below.

• d/dx (k) = 0, where k is any constant
• d/dx(x) = 1
• d/dx(xn) = nxn-1
• d/dx (kx) = k, where k is any constant
• d/dx (√x) = 1/2√x
• d/dx (1/x) = -1/x2
• d/dx (log x) = 1/x, x > 0
• d/dx (ex) = ex
• d/dx (ax) = ax ln a

## Types of Derivatives

Derivatives have been classified into various types based on their function.

### First-order derivatives

The first-order derivatives display the direction of the function. It shows us whether the function is increasing or decreasing and the first-order derivatives can be seen as an instantaneous rate of change. The first-order derivatives can also be predicted from the slope of a tangent line.

### Second-order derivatives

The second-order derivatives have graphical connotation and are used to derive the shape of the graph of a given function. The functions are classified based on their concavity into concave up and concave down. Let us take an example of a function f(x) which is equal to x.2

The derivative of x2 will be 2x which implies that for every unit change in x, the value of the function becomes twice the value. The derivative dx can also be made very small so that it becomes negligible. Limits is applied when x approaches zero but does not become zero. We can also say that for all the real values of ε > 0, there exists a real  δ > 0 such that for all x with 0 < |x − c| < δ, (here c ∈ R) we have |f(x) − L| < ε.

### Conclusion

Derivatives continue to remain essential while plotting various graphs, especially in business. Not only mathematics but derivatives are also used in physics, finance, and seismology. Derivative formulas are important while finding the tangent and normal to a curve, Newton’s method, and linear approximations.