Using the definition of derivative, find $f^{'} (1)$ for $f (x) = x + 3$ .

medium 2 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

f^{'} (1) = \frac{1}{4}

The correct answer is $f^{'} (1) = \frac{1}{4}$ — Thus $f^{'} (1) = \frac{1}{4}$ .

Step-by-step solution

Write the definition: $f^{'} (1) = lim_{h \to 0} \frac{( 1 + h ) + 3 - 1 + 3}{h}$ .
Simplify numerator by rationalising: $\frac{4 + h - 2}{h} \cdot \frac{4 + h + 2}{4 + h + 2} = \frac{( 4 + h ) - 4}{h ( 4 + h + 2 )}$ .
This reduces to $\frac{h}{h ( 4 + h + 2 )} = \frac{1}{4 + h + 2}$ .
Now let $h \to 0$ : $\frac{1}{4 + 2} = \frac{1}{2 + 2} = \frac{1}{4}$ .
Thus $f^{'} (1) = \frac{1}{4}$ .

Common mistake: Skipping rationalisation leading to indeterminate form

Concept tested: Derivative from first principles