Question

Find the derivative of f(x)=2√x + 5x - 3 using the formal definition.

Answer 1

\[\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}=\lim_{h \rightarrow 0}\frac{[2 \sqrt{x+h}+5(x+h)-3]-[2 \sqrt{x}+5 x -3]}{h}\]
\[=\lim_{h \rightarrow 0}\frac{2\sqrt{x+h}+5x+5h-3-2\sqrt{x}-5x+3}{h}\]
\[=\lim_{h \rightarrow 0}\frac{2\sqrt{x+h}-2\sqrt{x}+5h}{h}\]
\[=\lim_{h \rightarrow 0}\frac{2\sqrt{x+h}-2\sqrt{x}}{h} +\lim_{h \rightarrow 0}\frac{5h}{h}\]
\[=2\lim_{h \rightarrow 0}\frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}+\lim_{h \rightarrow 0}5\]
\[=2 \lim_{h \rightarrow 0}\frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})} +5\]
\[=2 \lim_{h \rightarrow 0}\frac{1}{\sqrt{x+h}+\sqrt{x}}+5\]
\[=2\cdot \frac{1}{\sqrt{x+0}+\sqrt{x}}+5=2 \cdot \frac{1}{2 \sqrt{x}}+5=\frac{1}{\sqrt{x}}+5\]