Find $f^{\prime }(2)$ for $f(x)=\sqrt{2x+5}$ using the definition of derivative.

Step-by-step solution

1. Write the definition: $f^{\prime }(2)={\lim }_{h\to 0}\frac{\sqrt{2(2+h)+5}-\sqrt{2\cdot 2+5}}{h}$.
2. Simplify the numerator: $\sqrt{2(2+h)+5}=\sqrt{9+2h}=\sqrt{9+2h}$.
3. Rationalise: multiply numerator and denominator by $\sqrt{9+2h}+3$ to obtain $\frac{(9+2h)-9}{h(\sqrt{9+2h}+3)}=\frac{2h}{h(\sqrt{9+2h}+3)}$.
4. Cancel $h$ (for $h\ne 0$): $\frac{2}{\sqrt{9+2h}+3}$.
5. Take the limit as $h\to 0$: $\frac{2}{\sqrt{9}+3}=\frac{2}{3+3}=\frac{1}{3}$.
6. Thus $f^{\prime }(2)=\frac{1}{3}$.

Concept tested: Derivative by definition

Published 12 September 2026 · How this solution was prepared and checked