Differentiation in calculus

The field of Calculus in mathematical sciences deals with change in functions. Differentiation in calculus is calculated to give information about the rate of change of one variable involved in the function, say x with respect to corresponding change of another variable in the same function, y. Differentiation is the building block of calculus. The method we use to find derivatives is called differentiation.

Calculus

Mathematics is a big field with subfields that focus on a particular part of it. Geometry deal with shapes and graphs, algebra deals with numbers, arithmetical operations, equations and inequalities. The same way Calculus is the field which deals with change

Differentiation

Differentiation is the process by which we find derivatives of a function. We can define derivative in the following manner:

Suppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by

lim_h→0[f(a+h)-f(a)]/h

It should be provided that this limit exists before attempting derivation. Derivative of f (x) at a is denoted by f′(a). f’(a) is the change in f(x) at a with respect to x

How to find a derivative of a function

It can be better understood with help of an example:

Find the derivative of f(x) = x2 + 7x  at x = 2.

The definition of derivative of a function is

lim_h→0[f(a+h)-f(a)]/h

We are to find the derivative at x=2

It can be written in the following way;

f’(x=2)= lim_h→0[f(2+h)-f(2)]/h= h0[{(2+h)²+ 7(2+h)}-{(2)²+7(2)}]/h

It can be simplified;

f’(2)= lim_h→0[4+h²+4h+14+7h-4-14]/h=  lim_{h→0[h²+11h]/h}

We can split the limit function into two parts

lim_h→0[h²/h] +lim_h→0[11h]/h

=0+11= 11

the derivative of f(x) = x² + 7x  at x = 2 comes out to be 11.

Some important results/rules of differentiation in calculus

These are the commonly used functions which are helpful in solving more complex problems of differentiation in calculus. All of these results can be proved easily by the definition of derivative of a function.

First we have the four algebraic operations and how they are used with differentiation of functions

1. Sum Rule: Derivative of the sum of two functions is the sum of the derivatives of the functions. It can be written as (u+v)’= u’+v’.
2. Difference Rule: Derivative of the difference of two functions is the difference of the derivatives of the functions. It can also be denoted like (u-v)’=u’-v’
3. Product Rule: Derivative of the product of two functions is given by the following product rule. It can be written in the form of: (uv)’=u’v+uv’
4. Quotient Rule: Derivative of two functions getting divided is given by the following quotient rule (whenever the denominator is non–zero). This rule is less straightforward than the others. Generally written as, (u/v)’ = (u’v-uv’)/v2

Then there are other rules that are also commonly used:

• Multiplication by constant: the constant brings no change to the derivation of the function except getting multiplied with the derivative to give the final result.

(cf)’ = cf’ where c is the constant and f is a function.

• Power Rule: 

f(x) = xn is nxn-1 for any positive integer n

• Reciprocal Rule: When we apply derivation on a reciprocal of a function then the derivative is negative one by function square times.

Can be represented mathematically like : (1/f)’ = -f/f2

• Chain Rule: Chain rule is used to calculate the derivation of the composite of two functions. It is one of the most important rules in derivation as highly complex functions can be solved using this.It states that the derivative of composite of two functions f and g result in the derivation, (fog)’ = f’(g(x))g’(x)

Differential equations

In calculus, differential equations are the equations containing functions with variables and their derivatives. Generally, in differential equations the functions represent the physical quantities, the derivatives in the equation will then represent the rate of change of these quantities and the differential equation themselves will represent a relationship between the physical quantities and the rate of change depending on variables present in that function/physical quantity.

Partial differentiation and partial equations

In case of differentiating one variable of a function is done by keeping the other variable as a constant then it is known as partial differentiation. Partial differentiation finds its applications in vectorial calculus. It is denoted as δx/δy where we are trying to find the rate of change of one variable, x when the other variable, y is assumed to be constant. Δ and δ both symbols can be used to represent partial differentiation in calculus.

Partial differential equations pose more complex than differential equations in general. Partial differential equations is an equation containing multiple partial derivatives of a function and the multiple variables of a function.

Partial differentiation in calculus and partial differential equations hold great importance in studying physics and various concepts of physics. It has direct application while calculating about quantities such as the propagation of heat or sound in different mediums(solid, liquid, gas and vacuum) , fluid flow of various liquids with different variables, elasticity, electrostatics and  electrodynamics

Conclusion

Calculus in mathematics deals with change in functions. Differentiation in calculus is calculated to give information about the rate of change of one variable with respect to another.Differential rules are rules in differentiation in calculus that are very commonly useful in solving more complex functions. All the differential rules can be derived from the differentiation method. To solve more complex questions we use them directly. These rules are mathematical operations of addition, subtraction, multiplication and division and their usage in algebra derivatives of functions, multiplication by constant, power rule, reciprocal rule and chain rule. When one variable of a function is done by keeping the other variable as a constant then it is known as partial differentiation.