Using the definition of derivative, find $f^{'} (5)$ for $f (x) = s q r t [3], 2 x + 1,$

medium 3 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

f^{'} (5) = \frac{2}{3 , 1 1 ^{2/3}}

The correct answer is $f^{'} (5) = \frac{2}{3 , 1 1 ^{2/3}}$ — 6. No simplification needed further.

Step-by-step solution

1. Write the difference quotient: $f^{'} (5) = lim_{h \to 0} \frac{3 2 ( 5 + h ) + 1 - 3 2 \cdot 5 + 1}{h}$
2. Simplify inside the radicals: $2 (5 + h) + 1 = 11 + 2 h$ , $2 c d o t 5 + 1 = 11$ .
3. Let $g (t) = t^{1/3}$ . Then $f^{'} (5) = lim_{h \to 0} \frac{g ( 11 + 2 h ) - g ( 11 )}{h}$ .
4. Use the known limit $d i s pl a y s t y l e l i m_{u \to 0} \frac{( a + u ) ^{1/3} - a ^{1/3}}{u} = \frac{1}{3 a ^{2/3}}$ with $a = 11$ , $u = 2 h$ . Hence numerator behaves like $\frac{1}{3 \cdot 1 1 ^{2/3}} \cdot 2 h$ for small $h$ .
5. Divide by $h$ : $f^{'} (5) = \frac{1}{3 \cdot 1 1 ^{2/3}} \cdot 2 = \frac{2}{3 , 1 1 ^{2/3}}$ .
6. No simplification needed further.

Common mistake: Applying the derivative formula for a cube root without accounting for the inner function

Concept tested: Derivative via definition

Published 14 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked