Question

Derivada de f(x) =2x^2–3/3X+2

Answer 1

Derivada de una división

La fórmula es
$\bf f(x)=\dfrac{M}{N} \quad\to\quad f'(x)= \dfrac{M'N-MN'}{N^{2}} \\\\\\Entonces\\\\\\ f(x)= \dfrac{2x^{2}-3}{3x+2} \quad\to\quad f'(x)= \dfrac{(2x^{2}-3)'(3x+2)-(2x^{2}-3)(3x+2)'}{(3x+2)^{2}}$

$\bf M= 2x^{2} -3\qquad M'=2x\\\\ N=3x+2\qquad\quad N'= 3\\\\\\ Reemplazamos \\\\\\ f'(x)= \dfrac{(2x^{2}-3)'(3x+2)-(2x^{2}-3)(3x+2)'}{(3x+2)^{2}} \\\\\\ f'(x)= \dfrac{(2x)(3x+2)-(2x^{2}-3)(3)}{(3x+2)^{2}} \qquad Resolvemos \\\\\\ f'(x)= \dfrac{6x^{2}+4x - (6x^{2} -9)}{(3x+2)^{2}} \\\\\\ f'(x)= \dfrac{6x^{2}+4x - 6x^{2} +9}{(3x+2)^{2}}$

$\bf f'(x)= \dfrac{4x +9}{(3x+2)^{2}} \\ \\ \\ \boxed{\bf f'(x)= \dfrac{4x +9}{(9x^{2} +12x+4)}}$

Espero que te sirva, salu2!!!!