Differentiate 2^² x w.r.t ² x.

(d)/(d(² x)) (2^² x ) = 2^² x 2

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 21(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section B, question 21(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(a) · 2 marks · Very short answerContinuity and Differentiability

Differentiate $2^{c o s^{2} x}$ w.r.t $cos^{2} x$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

Call the first function $u$ and the second $v$ . You need $\frac{d u}{d v}$ .

Hint 2

Find $\frac{d u}{d x}$ and $\frac{d v}{d x}$ separately, then divide them.

Hint 3

Remember that $\frac{d}{d x} a^{f (x)} = a^{f (x)} lo g a \cdot f^{'} (x)$ .

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 $u = 2^{c o s^{2} x}$ . By the chain rule, $\frac{d u}{d x} = 2^{c o s^{2} x} lo g 2 \cdot (- 2 cos x sin x)$ .
1 mark
Let $v = cos^{2} x$ . Then $\frac{d v}{d x} = - 2 cos x sin x$ .
½ mark
Divide the two derivatives: $\frac{d u}{d v} = \frac{d u / d x}{d v / d x} = \frac{2 ^{c o s^{2} x} lo g 2 ( - 2 cos x sin x )}{- 2 cos x sin x} = 2^{c o s^{2} x} lo g 2$ .
½ mark

Answer: $\frac{d}{d ( cos ^{2} x )} (2^{c o s^{2} x}) = 2^{c o s^{2} x} lo g 2$

Where the marks are

Correct derivative of $2^{c o s^{2} x}$ with respect to $x$ (1 mark)
Correct derivative of $cos^{2} x$ with respect to $x$ (½ mark)
Divides the two derivatives to get $2^{c o s^{2} x} lo g 2$ (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 paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 21(a)

Hints

Step-by-step solution

Revise the chapter

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