Find the derivative of 2 x-3-3 x-2 by first principle-

Let f(x) = 2x+3/3x+2 Then, f(x+h) = 2(x+h)+3/3(x+h)+2 ∴ f'(x) = lim_h → 02(x+h)+3/3(x+h)+2 2x+3/3x+2/h = lim_h → 0(2x+3+2h)(3x+2) (2x+3)(3x+2+3h)/h(3x+2)(3x+ ...

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Byju's Answer

Standard XI

Mathematics

Derivative from First Principle

Find the deri...

Question

Find the derivative of $\frac{2 x + 3}{3 x + 2}$ by first principle.

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Solution

Let f(x) = $\frac{2 x + 3}{3 x + 2}$

Then, $f (x + h) = \frac{2 (x + h) + 3}{3 (x + h) + 2}$

$∴ f^{'} (x) = {lim}_{h \to 0} \frac{\frac{2 (x + h) + 3}{3 (x + h) + 2} - \frac{2 x + 3}{3 x + 2}}{h}$

$= {lim}_{h \to 0} \frac{(2 x + 3 + 2 h) (3 x + 2) - (2 x + 3) (3 x + 2 + 3 h)}{h (3 x + 2) (3 x + 2 + 3 h)}$

$= {lim}_{h \to 0} \frac{(2 x + 3) (3 x + 2) + 2 h (3 x + 2) - (2 x + 3) (3 x + 2) - 3 h (2 x + 3)}{h (3 x + 2) (3 x + 2 + 3 h)}$

$= {lim}_{h \to 0} \frac{h (6 x + 4 - 6 x - 9)}{h (3 x + 2) (3 x + 2 + 3 h)} \Rightarrow f^{'} (x) = {lim}_{h \to 0} \frac{- 5}{(3 x + 2) (3 x + 2 + 3 h)}$

$= \frac{- 5}{(3 x + 2)^{2}}$

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