Derivatives

Calculus is also called infinitesimal calculus, or originally it was also called “the calculus of infinitesimals.” It is a mathematical study of continuous change. It is the same as geometry studies shape and algebra studies arithmetic operations. However, the fundamental tool of calculus is a derivative of a function. One can say that the measurement of sensitivity to change of output value and the input value is the derivative of a function. Calculus was developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century.

Definition of Derivative of Function - Calculus

Differential calculus and integral calculus are two major branches of calculus. The slope of curves and instantaneous rate of change is differential calculus, while the accumulation of quantities and areas under or between curves is integral calculus.

The process through which derivatives are found is known as differentiation. On the other hand, the reverse process for the same is called antidifferentiation. In a single variable calculus differentiation and integration are the two fundamental operations.

How to find a derivative of function y

A step by step process is followed to find the derivative.

We use the following  slope formula to find the derivatives of function :

Slope = \[\frac{Change in Y}{Change in X}\] = \[\frac{𝚫y}{𝚫x}\]

Thus we can say that x  changes to  x+Δx

Accordingly, y changes  to f(x+Δx)

Now if we expand the slope formula by replacing the values it will become as follows :

𝚫y = \[\frac{𝚫y}{𝚫x}\] = \[\frac{f(x+𝚫x) - f(x)}{𝚫x}\]

We would simplify the formula as best we can and at the end try to 𝚫x shrink towards zero.

Differentiation

Differentiation is important for finding rates of change. For example, differentiation helps us to find the rate of change of time for acceleration. Similarly, when on a graph of y against x, there is a gradient of the curve, then one can find the rate of change of x for y with the help of differentiation. For differentiating functions easily, there is a set of simple rules which can be followed.

Say if y is some expression having numbers in it and x then derivative of y can be written as dy/dx which is pronounced as  “dee y by dee x”

Also :

Say If y = xn, dy/dx = nxn-1

Again  If y = kxn, dy/dx = nkxn-1

In this “k” is some number

Thus we can see for differentiating x , the power is to be brought down to in front of x and then is reduced by one.

Notation

Different mathematicians have evolved many different notations for the derivative of function according to the context where they are used. Some of the common notations are as follows

• Leibniz’s notation

When a functional relationship between the dependent and independent variables y and x forms an equation y=f(x), then this notation is used. The derivative for the same is   

\[\frac{dy}{dx}\]

• Lagrange’s notation

A prime mark is used to express a derivative in Lagrange’ s notation. The prime mark for the first  derivative is

\[{f}'\](x),

Accordingly, the power goes on increasing for the further derivatives

• Euler’s Notation 

A differential operator is suggested in Euler’s notation. The derivative looks as follows when it is applied to a function of x:

(Df)(x) = \[\frac{df(x)}{dx}\]

• Newton's Notation

This notation is also called Dot notation. In this, a dot is placed on the dependent variable as follows

ẋ , ẍ  

• Partial derivatives

These notations are used for specific differentiation such as multivariate calculus or tensor analysis etc

• Notation in Vector calculus

Differentiation, as well as the integration of vector or scalar fields, are noted by vector calculus. The various symbols used in the differential operator are :

Gradient

∇φ

Divergence

∇∙A

Laplacian

∇2φ

Curl

∇×A

Properties of derivative and Chain Rule

Following are the properties of the derivative

• Linear Combination Rule

The linearity law is very important to emphasize its nature with alternate notation. Symbolically it is specified as 

\[{h}'\](x) = \[{af}'\](x) + \[{bg}'\](x)

• Product Rule

When a derivative of a product of 2 functions is to be found, the product Rule is used. It is represented as follows:

\[{[f(x)g(x)]}'\] = f(x)\[{g}'\] (x) + \[{f}'\](x)g(x)

• Quotient Rule

The derivative of one function divided by other is found by quotient rule such as :

            \[{\left [ \frac{f(x)}{g(x)} \right ]}'\]  = \[\frac{g(x){f}'(x) - f(x){g}'(x)}{[g(x)]^{2}}\]

• Chain Rule

To find the derivative of composite function or function of a function, one of the properties of derivative - Chain Rule is applied. When the given problem has chained dependencies, that is a study that involves the function of a second variable which depends on the function of the third quantity and so on.

The following are the two notations for the chain rule property of derivative:

              \[{f[g(x_{0})]}'\] = \[{f}'\][g(\[x_{0}\])]\[{g}'\] \[(x_{0})\]

              \[{fog}'\] (\[x_{0}\]) = [( \[{f}'\]og)(\[x_{0}\])] \[{g}'\]\[(x_{0})\] 

Conclusion

Thus derivative of function or calculus specifies an instant rate of change of a function at a certain point. Also, one can say that the derivative gives the slope of the line tangent to the function’s graph at a specific point. The derivative is important in many folds such as elementary calculus, multivariate calculus, and many others. The derivative explains not only the instant rate of change but also the acceleration, amongst other things. Derivatives are applied very effectively in real life even such as for infrastructure development, designing of buildings, bridges or in engineering concepts like electrical engineering, etc