What is the differential function of sqrt(x2 + 2) ?

Option b: x/sqrt(x2+2)dx. Solution: The differential of a function y = f(x) is defined as: dy = f'(x) dx Here, the function is: f(x) = sqrt(x2 + 2) = (x2 + 2)1/2 Let's find its derivative f'(x) using the chain rule: f'(x) = 1/2(x2 + 2)-1/2 × d/dx(x2 + 2) f'(x) = 1/2sqrt(x2 + 2) × (2x) f'(x) = x/sqrt(x2 + 2) Now, substitute this derivative into the expression for the differential: dy = x/sqrt(x2 + 2) dx Hence, Option B is the correct answer.

Basic Applications of CalculusMCQPYQ May 25Question 4039 of 32

All Questions

What is the differential function of $\displaystyle \sqrt{x^2 + 2}$ ?

Options

\displaystyle x\sqrt{x^2+2}dx

\displaystyle \frac{x}{\sqrt{x^2+2}}dx

\displaystyle \frac{x}{\sqrt{x^2-2}}dx

\displaystyle -\frac{x}{\sqrt{x^2+2}}dx

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Correct Answer

✅ Option b — $\displaystyle \frac{x}{\sqrt{x^2+2}}dx$

All Options:

A $\displaystyle x\sqrt{x^2+2}dx$
B $\displaystyle \frac{x}{\sqrt{x^2+2}}dx$
C $\displaystyle \frac{x}{\sqrt{x^2-2}}dx$
D $\displaystyle -\frac{x}{\sqrt{x^2+2}}dx$

Detailed Solution & Explanation

The differential of a function

\displaystyle y = f(x)

is defined as:

dy = f\'(x) dx

Here, the function is:

f(x) = \sqrt{x^2 + 2} = (x^2 + 2)^{1/2}

Let\'s find its derivative

\displaystyle f\'(x)

using the chain rule:

f\'(x) = \frac{1}{2}(x^2 + 2)^{-1/2} \times \frac{d}{dx}(x^2 + 2)

f\'(x) = \frac{1}{2\sqrt{x^2 + 2}} \times (2x)

f\'(x) = \frac{x}{\sqrt{x^2 + 2}}

Now, substitute this derivative into the expression for the differential:

dy = \frac{x}{\sqrt{x^2 + 2}} dx

Hence, **Option B** is the correct answer.

About This Chapter: Basic Applications of Calculus

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Limits, Continuity, Derivatives, Integrals

This chapter covers Limits, Continuity, Derivatives, Integrals and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

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Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Function

A relation from a set A (Domain) to a set B (Codomain) such that every element in A is associated with exactly one element in B. Written as f: A→B. Types include Linear, Quadratic, Polynomial, and Exponential functions.

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What is the differential function of x2+2\displaystyle \sqrt{x^2 + 2}x2+2​ ?