Differentiation in Calculus (Derivative Rules, Formulas, Solved Examples) (2024)

In calculus, differentiation is one of the two important concepts apart from integration. Differentiation is a method of finding the derivative of a function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables. The most common example is the rate change of displacement with respect to time, called velocity. The opposite of finding a derivative is anti-differentiation.

If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx. This is the general expression of derivative of a function and is represented as f'(x) = dy/dx, where y = f(x) is any function.

What is Differentiation in Maths

In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy / dx

If the function f(x) undergoes an infinitesimal change of ‘h’ near to any point ‘x’, then the derivative of the function is defined as

\(\begin{array}{l}\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\end{array} \)

Derivative of Function As Limits

If we are given with real valued function (f) and x is a point in its domain of definition, then the derivative of function, f, is given by:

f'(a) = lim_h→0[f(x + h) – f(x)]/h

provided this limit exists.

Let us see an example here for better understanding.

Example: Find the derivative of f(x) = 2x, at x =3.

Solution: By using the above formulas, we can find,

f'(3) =lim_h→0[f(3 + h) – f(3)]/h =lim_h→0[2(3 + h) – 2(3)]/h

f'(3) = lim_h→0[6 + 2h – 6]/h

f'(3) = lim_h→02h/h

f'(3) = lim_h→02 = 2

Also, checkContinuity And Differentiabilityto understand the above expression.

Notations

When a function is denoted as y = f(x), the derivative is indicated by the following notations.

1. D(y) or D[f(x)] is called Euler’s notation.
2. dy/dx is called Leibniz’s notation.
3. F’(x) is called Lagrange’s notation.

The meaning of differentiation is the process of determining the derivative of a function at any point.

Linear and Non-Linear Functions

Functions are generally classified into two categories under Calculus, namely:

(i) Linear functions

(ii) Non-linear functions

A linear function varies at a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in the case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a derivative of that particular function.

Differentiation Formulas

The important Differentiation formulasare given below in the table. Here, let us consider f(x) as a function and f'(x) is the derivative of the function.

Also, see:

Differentiation Rules

The basic differentiation rules that need to be followed are as follows:

• Sum and Difference Rule
• Product Rule
• Quotient Rule
• Chain Rule

Let us discuss all these rules here.

Sum or Difference Rule

If the function is the sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,

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

\(\begin{array}{l}f(x) = u(x) \times v(x)\end{array} \)

then,

\(\begin{array}{l}\mathbf { f'(x) = u'(x) \times v(x) + u(x) \times v'(x)}\end{array} \)

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,

\(\begin{array}{l}f(x) = \frac{u(x)}{v(x)}\end{array} \)

then,

\(\begin{array}{l}\large \mathbf { f'(x) = \frac{u'(x) \times v(x) – u(x) \times v'(x)}{(v(x))^{2}}}\end{array} \)

Chain Rule

If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as,

\(\begin{array}{l}\large \mathbf{\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} u} \times \frac{\mathrm{d}u }{\mathrm{d} x}}\end{array} \)

This plays a major role in the method of substitution that helps to perform differentiation of composite functions.

Real-Life Applications of Differentiation

With the help of differentiation, we are able to find the rate of change of one quantity with respect to another. Some of the examples are:

• Acceleration: Rate of change of velocity with respect to time
• To calculate the highest and lowest point of the curve in a graph or to know its turning point, the derivative function is used
• To find tangent and normal to a curve

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

This is the final answer.

Q.2: Differentiate y = x(3x² – 9)

Solution: Given, y = x(3x² – 9)

y = 3x³ – 9x

On differentiating both the sides we get,

dy/dx = 9x² – 9

This is the final answer.

Video Lesson on Class 12 Important Calculus Questions

Practice Problems

1. Find the derivative of the function f(x) = 3 sin x + cos x – tan x.
2. Perform the differentiation for the following functions:
   (i) f(x) = x³ sin 2x
   (ii) g(x) = 4xe^2x − 9x
3. Find the derivative of the function f(x) = x/(x – 2)
   (i) Using the limit definition of differentiation
   (ii) Using the quotient rule

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