Using the definition of derivative, find $f^{\prime }(3)$ for $f(x)=\sqrt{2x+1}$.

Step-by-step solution

1. Write the definition: $f^{\prime }(3)=\underset{h\to 0}{\lim }\frac{\sqrt{2(3+h)+1}-\sqrt{2\cdot 3+1}}{h}$.
2. Simplify inside the radicals: $\sqrt{2h+7}-\sqrt{7}$.
3. Multiply numerator and denominator by the conjugate $\sqrt{2h+7}+\sqrt{7}$ to rationalise: $\frac{2h}{h(\sqrt{2h+7}+\sqrt{7})}$.
4. Cancel the common factor $h$: $\frac{2}{\sqrt{2h+7}+\sqrt{7}}$.
5. Take the limit as $h\to 0$: the denominator becomes $2\sqrt{7}$.
6. Thus $f^{\prime }(3)=\frac{2}{2\sqrt{7}}=\frac{1}{\sqrt{7}}$.

Concept tested: Derivative definition for radical function

Published 19 September 2026 · How this solution was prepared and checked