If y=sin^-1(x^2+1x^4+3x^2+1), x>0, then dy/dx=

A. x^2-1x^4+3x^2+1. Step 1: x^4+3x^2+1=(x^2+1)^2+x^2, so with legs x^2+1 and x: siny=x^2+1x^4+3x^2+1 and tany=x^2+1x=x+1/x. Step 2: y=tan^-1(x+1/x), so dy/dx=1-1x^21+(x+1/x)^2. Step 3: Multiply top and bottom by x^2: x^2-1x^2+x^4+2x^2+1=x^2-1x^4+3x^2+1. Check: At x=1 the derivative is 0, as it must be: x+1/x has its minimum at x=1. Trap: (B) comes from the sign slip d/dx(x+1/x)=1+1x^2; it should be 1-1x^2.

Question 8 of 15 · Differentiation

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If $y = s in^{- 1} (\frac{x ^{2} + 1}{x ^{4} + 3 x ^{2} + 1})$ , $x > 0$ , then $\frac{d y}{d x} =$

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $\frac{x ^{2} - 1}{x ^{4} + 3 x ^{2} + 1}$

Detailed Solution

$x^{4} + 3 x^{2} + 1 = (x^{2} + 1)^{2} + x^{2}$ , so with legs $x^{2} + 1$ and $x$ : $s in y = \frac{x ^{2} + 1}{x ^{4} + 3 x ^{2} + 1}$ and $t an y = \frac{x ^{2} + 1}{x} = x + \frac{1}{x}$ .

$y = t an^{- 1} (x + \frac{1}{x})$ , so $\frac{d y}{d x} = \frac{1 - \frac{1}{x ^{2}}}{1 + ( x + \frac{1}{x} ) ^{2}}$ .

Multiply top and bottom by $x^{2}$ : $\frac{x ^{2} - 1}{x ^{2} + x ^{4} + 2 x ^{2} + 1} = \frac{x ^{2} - 1}{x ^{4} + 3 x ^{2} + 1}$ .

✓ Check

x = 1

the derivative is 0, as it must be:

x + \frac{1}{x}

has its minimum at

x = 1

⚠ Common trap

(B) comes from the sign slip

\frac{d}{d x} (x + \frac{1}{x}) = 1 + \frac{1}{x ^{2}}

; it should be

1 - \frac{1}{x ^{2}}

Quick tip

$(x^{2} + 1)^{2} + x^{2} = x^{4} + 3 x^{2} + 1 \Rightarrow y = tan^{- 1} (x + \frac{1}{x})$

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Differentiation — Preparation

This question tests your understanding of Differentiation, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

Differentiation finds the rate at which one quantity changes with respect to another. The derivative dy/dx gives the slope of a curve at any point. This topic covers derivatives of standard functions (powers, trigonometric, exponential and logarithmic functions), and the product, quotient and chain rules. It also covers derivatives of implicit functions, parametric functions, inverse trigonometric functions, logarithmic differentiation (for expressions like x^x), and higher-order derivatives. Exam questions often combine several rules in one problem, so identify the outermost function first and apply the chain rule step by step. For inverse trigonometric expressions, a clever substitution such as x = tan theta often simplifies the problem dramatically. Practise the differentiation questions below, each with a step-by-step solution showing which rule is applied.

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