Using the definition of derivative, find $f^{'} (2)$ for $f (x) = 3 x + 4$ .

easy 2 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

f^{'} (2) = \frac{3}{2 10}

The correct answer is $f^{'} (2) = \frac{3}{2 10}$ — Thus $f^{'} (2) = \frac{3}{2 10}$ .

Step-by-step solution

Write the definition: $f^{'} (2) = lim_{h \to 0} \frac{f ( 2 + h ) - f ( 2 )}{h}$ .
Substitute $f (x)$ : $\frac{3 ( 2 + h ) + 4 - 3 \cdot 2 + 4}{h} = \frac{6 + 3 h + 4 - 10}{h} = \frac{10 + 3 h - 10}{h}$ .
Rationalise the numerator by multiplying numerator and denominator by $10 + 3 h + 10$ .
The expression becomes $\frac{( 10 + 3 h ) - 10}{h ( 10 + 3 h + 10 )} = \frac{3 h}{h ( 10 + 3 h + 10 )}$ .
Cancel $h$ : $\frac{3}{10 + 3 h + 10}$ .
Take the limit $h \to 0$ : denominator $\to 210$ , so the limit is $\frac{3}{2 10}$ .
Thus $f^{'} (2) = \frac{3}{2 10}$ .

Common mistake: Forgetting to rationalise the numerator after substituting the function.

Concept tested: Derivative by definition