In today’s session we are going to discuss about the differential calculus. First of
all we must know about the calculus and its use. Calculus is a Latin word which is
a small stone that is used for counting.
Calculus is a field of mathematics. Calculus generally focuses on some topics of
mathematics like infinite series, integrals, limits, derivatives etc. The calculus is
classified into two broad sections, named as:
1. Differential Calculus
2. Integral Calculus.
These classifications of calculus are related to the fundamental theorem of
calculus. In calculus we study the change in any quantity or thing.
Today we are going to discuss about Differential Calculus. As mentioned above
that it is a subfield or category of the calculus, it relates with the study of the
changing rate of any quantity.
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In this subfield of calculus we learn about the derivative of functions. We learn to
find the derivative of a function at a given input value which further explains the
changing rate of the function near the given input value. This process of finding
the derivative of any function is known as Differentiation.
The fundamental theorem of the calculus tells us that the differentiation is just
reverse process of the integration.
Now the question arises is that what the Derivative is.
Derivatives: Assume that we have two real numbers which are y and z and z is the
function of y means that for each and every value of y there will be a
corresponding value for z too.
Then this relation between these two real numbers can be given as z = f(y); Now if
the function f(y) is the equation for any straight line then there also exists two real
numbers say M and B such that z = M*y + B where M is the slope of line and we
can determine this by:
M = change in z/change in y
M = dz/dy here d denotes the change in.
In above, if we take slope as a function f, then derivative of the function f at point
y is best approximation of the slope of function at point y. So the derivative of
f’(y) = dz/dy.
So this is the derivative of y.
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Differential calculus has several applications in real world; for instance, in physics
we learnt that the velocity of a body or an object is the derivative of the
displacement of the body or object in its moving state with respect to the time.
Similarly the derivation of the velocity of the moving body or object with respect
to the time is called acceleration.
One more example of the differential calculus is Newton’s Law of motion according
to which the force applied by a body is equal to the derivative of the momentum
of the same body. So this law is also using differential calculus.
To find maxima and minima of a function we get help of the derivatives in
mathematics. The equations which use the derivatives are called differential
Derivatives are used in almost all the fields of mathematics such complex
analysis, function analysis, measure theory, differential geometry etc.
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