The concept of differentiation is one of the two most important concepts in calculus, along with integration. Calculating a function's derivative is called differentiation. Using one of a function's variables, differentiation determines the instantaneous rate at which it changes.
One of the most common examples is the change in displacement over time, known as velocity. A derivative cannot be found if an anti-differentiation is performed.
If x is a variable and y is another variable, then dy/dx represents the rate of change of x with respect to y. The derivative of a function can be expressed as f'(x) = dy/dx, where y = f(x) is any function.
In mathematics, what is differentiation?
Differentiation is defined in mathematics as the derivative of a function based on an independent variable. An independent variable can be changed by one unit by using differentiation in calculus.
Suppose y = f(x) is a function of x. In this case, the rate of change of "y" per unit change in "x" is as follows: dy / dx
Near any point 'x’ if f(x) undergoes a small change in 'h,' then the derivative is
Functions as limits of derivatives:
If x belongs to the domain of definition of a real-valued function (f), then we can predict the derivative by:
f'(a) = limh→0[f(x + h) – f(x)]/h
provided this limit exists.
Let us see an example here for a better understanding.
Example: Find the derivative of f(x) = 2x, at x =3.
Solution:
By using the above formulas, we can find,
f'(3) = limh→0 [f(3 + h) – f(3)]/h = limh→0[2(3 + h) – 2(3)]/h
f'(3) = limh→0 [6 + 2h – 6]/h
f'(3) = limh→0 2h/h
f'(3) = limh→0 2 = 2
Notations:
When a function is denoted as y = f(x), the derivative is indicated by the following notations.
D(y) or D[f(x)] is called Euler’s notation.
dy/dx is called Leibniz’s notation.
F’(x) is called Lagrange’s notation.
In mathematics, differentiation refers to determining an object's derivative at any point in time.
Linear and Non-Linear Functions:
Functions are generally classified into two categories under Calculus, namely:
(i) Linear functions
(ii) Non-linear function
Through its domain, a linear function varies at a constant rate. As a result, the overall rate of change of a function is the same as the rate of change at any particular point.
In the case of non-linear functions, the rate of change varies from point to point. The variation depends on the nature of the function.
A derivative of a function is the rate of change of that function at a given point.
Differentiation Formulas:
The important Differentiation formulas are given below in the table. Here, let us consider f(x) as a function, and f'(x) is the derivative of the function.
· If f(x) = tan (x), then f'(x) = sec2x
· If f(x) = cos (x), then f'(x) = -sin x
· If f(x) = sin (x), then f'(x) = cos x
· If f(x) = ln(x), then f'(x) = 1/x
· If f(x) = ex, then f'(x) = ex
· If f(x) = xn, where n is any fraction or integer, then f'(x) = nxn-1
· If f(x) = k, where k is a constant, then f'(x) = 0
Rules of Differentiation:
Following are the basic differentiation rules:
Sum and Difference Rule
Product Rule
Quotient Rule
Chain Rule
Here are all the rules we need to discuss.
Rule of Sum or Difference:
If a function is the sum or difference of two functions, the derivative of the function is the sum or difference of the individual functions.
If f(x) = u(x) ± v(x)
then, f'(x) = u'(x) ± v'(x)
Product Rule:
As per the product rule, if the function f(x) is product of two functions u(x) and v(x), the derivative of the function is,
If
Quotient rule:
If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is
If,
Chain Rule:
If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as,
It plays an important role in the substitution method, which helps to perform the differentiation of complex functions
Practical application of differentiation:
Differentiation can be used to find the rate of change of one quantity with respect to another quantity. Some of the examples are:
acceleration: rate of change of velocity over time
Differential functions are used to calculate the highest and lowest points of a curve on a graph and to find its turning points
find tangents and normal of curves
Solved Examples:
Q.1: Differentiate f(x) = 6x3 – 9x + 4 with respect to x. Solution: Given: f(x) = 6x3 – 9x + 4
On differentiating both the sides w.r.t x, we get;
f'(x) = (3)(6)x2 – 9
f'(x) = 18x2 – 9
Some Sample Lectures on DIFFERENTIATION by Y.S Sir:
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