Differentiate ((5^x)/(x^5) ) with respect to x.

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 23(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, 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

Differentiate $(\frac{5 ^{x}}{x ^{5}})$ with respect to x.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

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

Hint 1

Rewrite the quotient as $5^{x} \cdot x^{- 5}$ .

Hint 2

Recall that $\frac{d}{d x} (a^{x}) = a^{x} lo g a$ .

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.

Write $y = 5^{x} \cdot x^{- 5}$ . By the product rule, $\frac{d y}{d x} = (5^{x})^{'} x^{- 5} + 5^{x} (x^{- 5})^{'} = 5^{x} lo g 5 \cdot x^{- 5} + 5^{x} (- 5 x^{- 6})$ .
1 mark
$\frac{d y}{d x} = \frac{5 ^{x} l o g 5}{x ^{5}} - \frac{5 ^{x + 1}}{x ^{6}}$
1 mark

Answer: $\frac{d y}{d x} = \frac{5 ^{x} l o g 5}{x ^{5}} - \frac{5 ^{x + 1}}{x ^{6}}$

Where the marks are

Applies the product (or quotient) rule with $\frac{d}{d x} 5^{x} = 5^{x} lo g 5$ (1 mark)
Correct simplified derivative (1 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 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 23(a)

Hints

Step-by-step solution

Revise the chapter

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