📘 Algebra of Derivative of Functions – Class 11 Mathematics
Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS
🔍 Introduction
In Class 11 Mathematics, derivatives play a crucial role in understanding how functions change. The algebra of derivatives provides a set of rules that help simplify the process of finding derivatives of complex functions using basic algebraic operations like addition, subtraction, multiplication, and division.
These rules are particularly important for students preparing for CBSE Board Exams, JEE Mains, and JEE Advanced.
Derivatives are defined as the varying rate of change of a function with respect to an independent variable. A derivative in calculus is the rate of change of a quantity y with respect to another quantity x. It is also termed the differential coefficient of y with respect to x. Differentiation is the process of finding the derivative of a function.
Table of Contents
How to Find Derivative of Function f(x) ?
Derivative of a function f(x) signifies the rate of change of the function f(x) with respect to x at a point lying in its domain.
Let f is a real-valued function and ‘a’ is any point in its domain for which f is defined then f(x) is said to be differentiable at the point x=a if the derivative f'(a) exists at every point in its domain. It is given by
\(\begin{array}{l}f'(a) = \displaystyle{\lim_{h \to 0} \frac{f(a+h)-f(a)}{(a+h)-a}}\end{array} \)
or
\(\begin{array}{l}f'(a) = \displaystyle{\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}}\end{array} \)
Given that this limit exists and f’(a) represents the derivative of f(x) at a. This is the first principle of the derivative.
The domain of f’(a) is defined by the existence of its limits. The derivative is also denoted as
\(\begin{array}{l}\frac{d}{dx}, f(x) \;\; or \;\; D(f(x))\end{array} \).
If y = f(x) then derivative of f(x) is given as \(\begin{array}{l}\frac{\mathrm{d} }{\mathrm{d} x}\end{array} \)or y’.
This is known as a derivative of y with respect to x.
Also, the derivative of a function f in x at x = a is given as:
\(\begin{array}{l} \frac{\mathrm{d} }{\mathrm{d} x} f(x)|_{x = a}\end{array} \)
or
\(\begin{array}{l} \frac{\mathrm{d} f}{\mathrm{d} x} |_{x = a}\end{array} \)
For a function to be differentiable at any point x = a in its domain, it must be continuous at that particular point but vice-versa is necessarily not always true.
Algebra or Rules of Derivatives of Functions
Suppose f(x) and g(x) are two functions such that their derivatives are defined in a common domain. Then we can define the following rules for the functions f and g.
Sum Rule of Differentiation
The derivative of the sum of two functions is the sum of the derivatives of the functions. This can be expressed as:
| \(\begin{array}{l}\large \mathbf{\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)}\end{array} \) |
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Difference Rule of Differentiation
The derivative of the difference of two functions is the difference of the derivatives of the functions. This can be expressed as:
| \(\begin{array}{l}\large \mathbf{\frac{d}{dx}[f(x)-g(x)]=\frac{d}{dx}f(x)-\frac{d}{dx}g(x)}\end{array} \) |
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Product Rule of Differentiation
The derivative of product of two functions f(x) and g(x) is given by the formula:
| \(\begin{array}{l}\large \mathbf{\frac{d}{dx}[f(x).g(x)]=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)}\end{array} \) |
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Quotient or Division Rule of Differentiation
The following formula gives the derivative of the quotient of two functions, provided the denominator is non–zero.
| \(\begin{array}{l}\large \mathbf{\frac{d}{dx}\left ( \frac{f(x)}{g(x)} \right )=\frac{g(x).\frac{d}{dx}f(x)-f(x).\frac{d}{dx}g(x)}{[g(x)]^2}}\end{array} \) |
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The derivatives of product and quotient of two functions can also be expressed as given below.
Consider u = f(x) and v = g(x), then the product of these two functions can be written as:
| (uv)′ = uv′ + u′v |
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Also quotient of two functions can be written as:
| (u/v)′ = (u′v – uv′)/v2 |
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These are referred to as a Leibnitz rule for differentiating the product and quotient of functions respectively.
Chain Rule:
Whenever a quantity ‘y’ varies with another quantity ‘x’ such that y = f(x), then f’(x) indicates the rate of change of y with respect to x (at x = x0).
Also, if two variables ‘x’ and ‘y’ are varying with respect to a third variable say ‘t’ then according to the chain rule, we have
| \(\begin{array}{l}\large \left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right ) = \frac{\frac{\mathrm{d} y}{\mathrm{d} t}}{\frac{\mathrm{d} t}{\mathrm{d} x}},\;\; where \;\; \frac{\mathrm{d} t}{\mathrm{d} x}\neq 0\end{array} \) |
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Constant Multiple Rule
If c is a constant and f(x) is a differentiable function, then:
\begin{array}{l} \mathbf{ \frac{d}{dx} \left[ c \cdot f(x) \right] = c \cdot \frac{d}{dx} f(x) } \end{array}
Explanation: A constant can be taken outside the derivative operation.
Derivatives of Some Basic Functions
The table below shows the derivatives of some standard basic functions.
| Derivative of a function | Result |
| d(x)/dx | 1 |
| d(ax)/dx | a |
| d(xn)/dx | nxn-1 |
| d(sin x)/dx | cos x |
| d(cos x)/dx | -sin x |
| d(tan x)/dx | sec2x |
| d(cosec x)/dx | -cosec x cot x |
| d(sec x)/dx | sec x tan x |
| d(cot x)/dx | -cosec2x |
| d(lnx)/dx | 1/x |
| d(ex)/dx | ex |
| d(ax)dx | ax (log a) |
Solved Examples on Algebra of Derivative of Functions
Example 1: Find the derivative of the function x + (1/x).
Solution:
Given: x + (1/x)
Let f(x) = x and g(x) = 1/x
Using the sum rule of differentiation,
d/dx [f(x) + g(x)] = d/dx f(x) + d/dx g(x)
d/dx [x + (1/x)] = d/dx (x) + d/dx (1/x)
= 1 + (-1/x2)
= 1 – (1/x2)
Example 2: Find the derivative of sin x – cos x.
Solution:
Given function is: sin x – cos x
Let f(x) = sin x and g(x) = cos x
Using the difference rule of differentiation,
d/dx [f(x) – g(x)] = d/dx f(x) – d/dx g(x)
d/dx (sin x – cos x) = d/dx (sin x) – d/dx (cos x)
= cos x – (-sin x)
= cos x + sin x
Example 3: Find the derivative of (5x3 – 3x + 1)(x + 1).
Solution:
Given: (5x3 – 3x + 1)(x + 1)
Let f(x) = (5x3 – 3x + 1) and g(x) = (x + 1)
Using the product rule of differentiation,
d/dx [f(x) .g(x)] = f(x) [d/dx g(x)] + g(x) [d/dx f(x)]
= (5x3 – 3x + 1) [d/dx (x + 1)] + (x + 1) [d/dx (5x3 – 3x + 1)]
= (5x3 – 3x + 1) (1 + 0) + (x + 1)[5(3x2) – 3(1) + 0]
= (5x3 – 3x + 1) + (x + 1)(15×2 – 3)
= 5x3 – 3x + 1 + 15x3 – 3x + 15x2 – 3
= 20x3 + 15x2 – 6x – 2
Example 4: Compute the derivative of f(x) = cot x.
Solution:
Given,
f(x) = cot x
This can be written as: f(x) = cos x/sin x
Let u(x) = cos x and v(x) = sin x
Using quotient rule or Leibnitz rule of quotient,
d/dx f(x) = d/dx [u/v]
= (u/v)′
= (u′v – uv′)/v2
= {[d/dx (cos x)] sin x – cos x[d/dx (sin x)]} / sin2x
= [(-sin x) sin x – cos x (cos x)]/ sin2x
= -[sin2x + cos2x]/ sin2x
= -1/sin2x
= -cosec2x
Example.5: Determine the rate of change of the volume of a sphere with respect to its radius ‘r’ when r = 3 cm.
Solution: We know the Volume of a Sphere is given as
V = \(\begin{array}{l}\frac{4}{3} \pi r^{3}\end{array} \).
Rate of change of Volume w.r.t. Radius is given as
\(\begin{array}{l}\frac{\mathrm{d} V}{\mathrm{d} r} = \frac{\mathrm{d} }{\mathrm{d} r}\left ( \frac{4}{3} \pi r^{3} \right ) = \frac{4}{3} \times \pi (3r^{2})\end{array} \)
\(\begin{array}{l}\frac{\mathrm{d} V}{\mathrm{d} r} = 4 \pi r^{2}\end{array} \)
When r = 3 cm
dV/dr = 36Π cm3/sec
Practice Questions on Algebra of Derivative of Functions
1. If f(x) = 3x² – 2x + 5 and g(x) = x³ + 4x, find the derivative of h(x) = f(x) + g(x).
2. Given f(x) = x² and g(x) = sin(x), find the derivative of their product: (f·g)(x).
3. If f(x) = ex and g(x) = ln(x), determine the derivative of their composition: f(g(x)).
4. Find the derivative of h(x) = (x² + 1) / (x – 2) using the quotient rule.
5. If f(x) = x³ and g(x) = cos(x), find the derivative of their difference: f(x) – g(x).
FAQs on Algebra of Derivative of Functions
What is a derivative of a function?
The derivative of a function represents the rate at which the function’s value changes with respect to its input variable. Geometrically, it represents the slope of the tangent line to the graph of the function at a given point.
What is the algebra of derivatives?
The algebra of derivatives refers to the rules and properties that govern the differentiation process. These rules allow us to find the derivatives of more complex functions by applying derivative rules to simpler functions.
What are the basic derivative rules?
Some basic derivative rules include the power rule, product rule, quotient rule, chain rule, and sum/difference rule. These rules provide a systematic way to find the derivatives of various types of functions.
When do we use the chain rule?
The chain rule is used to find the derivative of a composite function, where one function is applied to the output of another function.
🔑 Why is the Algebra of Derivatives Important?
- Simplifies the process of differentiating complex functions.
- Helps solve real-world problems involving rates of change, velocity, and optimization.
- Essential for solving higher-level calculus problems in competitive exams like JEE Mains and Advanced.
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