If f(x+y)=f(x)f(y) for all x,y R and f(5)=2, f'(0)=3, then using the definition of derivatives, find f'(5).

MathsCBSE Class 12 · Sample paper, session 2025-26 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 24

From the official CBSE sample question paper for 2025-26, section B, question 24, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24 · 2 marks · Very short answerContinuity and Differentiability

If $f (x + y) = f (x) f (y)$ for all $x, y \in R$ and $f (5) = 2$ , $f^{'} (0) = 3$ , then using the definition of derivatives, find $f^{'} (5)$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Substitute $x = 0$ in the functional equation to find $f (0)$ .

Hint 2

Write $f^{'} (5)$ by first principles and replace $f (5 + h)$ by $f (5) f (h)$ .

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.

Put $x = 0, y = 5$ : $f (5) = f (0) f (5) \Rightarrow 2 = 2 f (0) \Rightarrow f (0) = 1$ .
½ mark
By definition, $f^{'} (5) = lim_{h \to 0} \frac{f ( 5 + h ) - f ( 5 )}{h} = lim_{h \to 0} \frac{f ( 5 ) f ( h ) - f ( 5 )}{h}$ [ $f (x + y) = f (x) f (y)$ ] $= lim_{h \to 0} \frac{2 f ( h ) - 2}{h} = 2 lim_{h \to 0} \frac{f ( h ) - f ( 0 )}{h}$ [ $f (0) = 1$ ].
1 mark
$= 2 f^{'} (0) = 2 \times 3 = 6$ .
½ mark

Answer: $f^{'} (5) = 6$

Where the marks are

Finds $f (0) = 1$ (½ mark)
Uses first-principle definition and functional equation to reach $2 lim \frac{f ( h ) - f ( 0 )}{h}$ (1 mark)
Gets $f^{'} (5) = 2 f^{'} (0) = 6$ (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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 24

Hints

Step-by-step solution

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