Differentiation Important Questions
Differentiation important questions with detailed solutions and answers are provided for students of Class 11 and Class 12 at Vedantu. Differentiation is a significant topic for Class 11th and 12th students since these concepts are further included in higher studies. Keeping in mind the importance of the topic, the problems prepared in differentiation of function worksheets are as per the CBSE board and NCERT curriculum. Practicing these questions will certainly aid students to solve tough problems and to score higher marks in the examination.
Differentiation of Function
The differentiation of a function f(x) is denoted in the form of f’(x). It implies that if f(x) = y, then f’(x) = dy/dx, which indicates y is differentiated in reference to x. Before we begin solving some questions based on differentiation, let us look into the general differentiation formulas used in this chapter.
Formulas in Differentiation of Function
Following are the important formulas used in solving the differentiation questions
Identifying the derivative of a function from first principles needs a lengthy calculation and it becomes quite possible to make mistakes. However, we can use certain methods of finding the derivative from first principles in order to obtain rules which make finding the derivative of a function much simpler.
Below mentioned are the rules of differentiation that are applied to obtain the derivative and differentiate with respect to.
Rules of Differentiation
Solved Examples on Differentiation Questions with Answers
Example: Differentiate (a2 - x2 / a2 + x2) ½ with respect to x.
Solution: Given, y = (a2 - x2 / a2 + x2)…√
Taking log on both the sides, we have
Log y = ½ log (a2 - x2 / a2 + x2)
Log y = ½ [log (a2 - x2) – log (a2 + x2)
Log y = ½ [log (a2 - x2) – ½ log (a2 + x2)
1/y dy/dx = ½ * 1/ (a2 - x2) x (-2x)-1/2 * 2x / a2 + x2
1/y dy/dx = -x /a2 - x2 -x/ a2 + x2
dy/dx = y -x /a2 [- x(1/ a2 - x2 +1/ a2 + x2)]
dy/dx = a2 - x2 / a2 + x2…√ [-x(2a2/ a2 - x2]
Example: Differentiate x5 wrt x.
Solution: Given, y = x5
On differentiating with respect to x, we get;
dy/dx = d(x5)/dx
y’ = 5x5-1
= 5x4
Hence, d(x5)/dx = 5x4
Example: Differentiate 10x2 wrt x.
Solution: Given, y = 10x2
y’ = d(10x2)/dx
y’ = 2.10.x = 20x
Hence, d(10x2)/dx = 20 x
Example: Differentiate the function 20x-4 + 9.
Solution: Given; y = 20x-4 + 9
y’ = d(20x-4 + 9)/dx
y’ = d(20x-4)/dx + d(9)/dx
y’ = -4.20.x-4-1+0
y’ = -80x-5
Hence, d (20x-4 + 9)/dx = -80x-5
Example: Differentiate the function of sin (3x+5)
Solution: Let’s say, y = sin (3x+5)
dy/dx = d[sin(3x+5)]/dx
= cos (3x+5) d (3x+5)/dx [Using the chain rule]
= cos (3x+5) [3]
Y’ = 3 cos (3x+5)
d [sin(3x+5)]/dx = 3 cos (3x+5)
Fun Facts
The opposite of determining a derivative is what we call anti-differentiation
Let’s say x is a variable and y is another variable, then the rate of change of x with respect to y is mathematically expressed as dy/dx. This is a usual expression of derivative of a function and is denoted as f'(x) = dy/dx, where y = f(x) is any function.
FAQs on Differentiation Questions
Q1. What is Meant by Differentiation?
Answer: In mathematics, the process of determining the derivative of a function is known as differentiation. It is actually one of the two important concepts apart from integration in calculus. The required derivative of a function can also be worked out by taking the limit of the difference quotient. However, it would not just be tedious but tiresome to have to do this every time you wanted to find the derivative of a function, for there are different rules of differentiation. These rules enable us to identify the derivative of a desired function directly. Thus, it is important for the students to equip themselves with the basic rules of differentiation in order to be able to apply them in topics like marginal analysis, optimization (Unconstrained and Constrained) problems.
Q2. What Problems Can I Solve with Basic Differentiation Questions?
Answer: Differentiation is a technique of identifying the derivative of a function, which further helps us in:
Finding the instantaneous rate of change in function depending upon one of its variables.
Finding the rate change of displacement with respect to time, known as velocity.
measuring the lowest and highest point of the curve in a graph or to find out its turning point
To identify tangent and normal to a curve
Q3. What is Meant by Linear and Nonlinear Functions?
Answer: In calculus, functions are usually classified into:
(i) Linear functions
(ii) Non-linear functions
A linear function differs with a constant rate through its domain. Thus, the overall rate of change of the function remains the same as the rate of change of a function at any point.
On the other hand, the rate of change of function differs from point to point in case of non-linear functions.
