Find the derivative of ^-1x with respect to x; (where x(1,∞)).

MathsCBSE Class 12 · Sample paper, session 2024-25 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 23(a)

From the official CBSE sample question paper for 2024-25, section B, question 23(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(a) · 2 marks · Very short answerContinuity and Differentiability

Find the derivative of $tan^{- 1} x$ with respect to $lo g x$ ; (where $x \in (1, \infty)$ ).

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

Open one hint at a time and try again before the next.

Hint 1

Call the two functions $y$ and $z$ and differentiate each with respect to $x$ .

Hint 2

The derivative of $y$ with respect to $z$ is the ratio of the two derivatives.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let $y = tan^{- 1} x$ and $z = lo g x$ . Then $\frac{d y}{d x} = \frac{1}{1 + x ^{2}}$ .
½ mark
$\frac{d z}{d x} = \frac{1}{x}$ .
½ mark
$\frac{d y}{d z} = \frac{d y / d x}{d z / d x}$ .
½ mark
$\frac{d y}{d z} = \frac{1/ ( 1 + x ^{2} )}{1/ x} = \frac{x}{1 + x ^{2}}$ .
½ mark

Answer: $\frac{x}{1 + x ^{2}}$

Where the marks are

$\frac{d y}{d x} = \frac{1}{1 + x ^{2}}$ (½ mark)
$\frac{d z}{d x} = \frac{1}{x}$ (½ mark)
Uses $\frac{d y}{d z} = \frac{d y / d x}{d z / d x}$ (½ mark)
Gets $\frac{x}{1 + x ^{2}}$ (½ mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 23(a)

Hints

Step-by-step solution

Revise the chapter

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