Logan Carlson
02/27/2024 · Junior High School
4. Calcula el diferencial de la función: \( y=\frac{3 x+7}{2 x+9} \)
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Step-by-step Solution
Para calcular el diferencial de la función \( y = \frac{3x + 7}{2x + 9} \), primero necesitamos encontrar la derivada de \( y \) con respecto a \( x \).
Usaremos la regla del cociente, que establece que si \( y = \frac{u}{v} \), entonces:
\[
\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]
En nuestro caso, tenemos:
- \( u = 3x + 7 \) y \( v = 2x + 9 \)
Calculamos las derivadas \( \frac{du}{dx} \) y \( \frac{dv}{dx} \):
\[
\frac{du}{dx} = 3
\]
\[
\frac{dv}{dx} = 2
\]
Ahora aplicamos la regla del cociente:
\[
\frac{dy}{dx} = \frac{(2x + 9)(3) - (3x + 7)(2)}{(2x + 9)^2}
\]
Simplificamos el numerador:
\[
= \frac{(6x + 27) - (6x + 14)}{(2x + 9)^2}
\]
\[
= \frac{6x + 27 - 6x - 14}{(2x + 9)^2}
\]
\[
= \frac{13}{(2x + 9)^2}
\]
Por lo tanto, la derivada es:
\[
\frac{dy}{dx} = \frac{13}{(2x + 9)^2}
\]
Ahora, el diferencial \( dy \) se puede expresar como:
\[
dy = \frac{dy}{dx} \cdot dx = \frac{13}{(2x + 9)^2} \cdot dx
\]
Así que el diferencial de la función \( y = \frac{3x + 7}{2x + 9} \) es:
\[
dy = \frac{13}{(2x + 9)^2} \, dx
\]
Quick Answer
El diferencial de la función \( y = \frac{3x + 7}{2x + 9} \) es \( dy = \frac{13}{(2x + 9)^2} \, dx \).
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