From the first principle find derivative of $\displaystyle \frac{3x+5}{\sqrt{x}}.$
I could successfully find the derivative using various rules but I get stuck when I try to solve it using first principle, please help me find derivative of the following using first principle.
Hint. We have that $$f(x)=\frac{3x+5}{\sqrt{x}}=3\sqrt{x}+\frac{5}{\sqrt{x}}.$$ Hence $$\begin{align}\frac{f(x+h)-f(x)}{h}&=3\frac{\sqrt{x+h}-\sqrt{x}}{h}+\frac{5}{h}\left(\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}\right)\\ &=\left(3-\frac{5}{{\sqrt{x}\sqrt{x+h}}}\right)\cdot\frac{\sqrt{x+h}-\sqrt{x}}{h}\end{align}$$ Now multiply the numerator and the denominator of the fraction on the right by $\sqrt{x+h}+\sqrt{x}$. Can you take it from here?