Find the differential of x^ x + (2x² - 3)/(2x² - x + 2) with respect to x.

x^ x (( x)/(x)-cosec²x x )+(-2x²+20x-3)/((2x²-x+2)²)

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2026: question 33(b)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section D, question 33(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(b) · 5 marks · Long answerContinuity and Differentiability

Find the differential of $x^{c o t x} + \frac{2 x ^{2} - 3}{2 x ^{2} - x + 2}$ with respect to x.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

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

Hint 1

Differentiate the two terms separately.

Hint 2

For $x^{c o t x}$ , take logs first and then use the product rule.

Hint 3

For the rational term, use the quotient rule and expand the numerator carefully.

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 $u = x^{c o t x}$ and $v = \frac{2 x ^{2} - 3}{2 x ^{2} - x + 2}$ , so the required derivative is $\frac{d u}{d x} + \frac{d v}{d x}$ .
½ mark
$lo g u = cot x \cdot lo g x$
½ mark
$\frac{1}{u} \frac{d u}{d x} = \frac{c o t x}{x} + lo g x (- cosec^{2} x)$
1 mark
$\frac{d u}{d x} = x^{c o t x} (\frac{c o t x}{x} - cosec^{2} x lo g x)$
½ mark
By the quotient rule, $\frac{d v}{d x} = \frac{( 2 x ^{2} - x + 2 ) ( 4 x ) - ( 2 x ^{2} - 3 ) ( 4 x - 1 )}{( 2 x ^{2} - x + 2 ) ^{2}}$
1 mark
Numerator $= (8 x^{3} - 4 x^{2} + 8 x) - (8 x^{3} - 2 x^{2} - 12 x + 3) = - 2 x^{2} + 20 x - 3$ , so $\frac{d v}{d x} = \frac{- 2 x ^{2} + 20 x - 3}{( 2 x ^{2} - x + 2 ) ^{2}}$
1 mark
Required derivative $= x^{c o t x} (\frac{c o t x}{x} - cosec^{2} x lo g x) + \frac{- 2 x ^{2} + 20 x - 3}{( 2 x ^{2} - x + 2 ) ^{2}}$
½ mark

Answer: $x^{c o t x} (\frac{c o t x}{x} - cosec^{2} x lo g x) + \frac{- 2 x ^{2} + 20 x - 3}{( 2 x ^{2} - x + 2 ) ^{2}}$

Where the marks are

Splits into $u$ and $v$ (½ mark)
Takes log: $lo g u = cot x lo g x$ (½ mark)
Differentiates $lo g u$ correctly (1 mark)
Correct $\frac{d u}{d x}$ (½ mark)
Applies the quotient rule to $v$ (1 mark)
Simplifies the numerator to $- 2 x^{2} + 20 x - 3$ (1 mark)
Adds the two results (½ mark)

Revise the chapter

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

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 paper2026 board paper, set 65/5/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2026: question 33(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Continuity and Differentiability