If-f-x-2x-3-prove-that-f-x-1-2x-3-

Question Number 217664 by hardmath last updated on 17/Mar/25

If f (x) = \sqrt{2 x + 3}

prove that : f^{'} (x) = \frac{1}{\sqrt{2 x + 3}}

Answered by SdC355 last updated on 18/Mar/25

f^{(1)} (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} .

\frac{d}{d x} \sqrt{2 x + 3} = \lim_{h \to 0} \frac{\sqrt{2 (x + h) + 3} - \sqrt{2 x + 3}}{h}

\lim_{h \to 0} \frac{\sqrt{2 x + 2 h + 3} - \sqrt{2 x + 3}}{h} =

\lim_{h \to 0} \frac{(\sqrt{2 x + 2 h + 3} - \sqrt{2 x + 3})}{h} \cdot \frac{(\sqrt{2 x + 2 h + 3} + \sqrt{2 x + 3})}{(\sqrt{2 x + 2 h + 3} + \sqrt{2 x + 3})}

∴ \lim_{h \to 0} \frac{2 x + 2 h + 3 - (2 x + 3)}{h (\sqrt{2 x + 2 h + 3} + \sqrt{2 x + 3})}

\lim_{h \to 0} \frac{2 x + 2 h + 3 - 2 x - 3}{h (\sqrt{2 x + 2 h + 3} + \sqrt{2 x + 3})} =

\lim_{h \to 0} \frac{2 h}{h (\sqrt{2 x + h + 3} + \sqrt{2 x + 3}} = \frac{2}{2 \sqrt{2 x + 3}}

\frac{1}{\sqrt{2 x + 3}}

Commented by hardmath last updated on 21/Mar/25

thankyou dearSir