Pregunta
Use the Quotient Rule to evaluate and simplify \( \frac{d}{d x}\left(\frac{x-4}{5 x-2}\right) \). \( \frac{d}{d x}\left(\frac{x-4}{5 x-2}\right)=\square \)
Ask by Rogers Bush. in the United States
Mar 09,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\[
\frac{d}{dx}\left(\frac{x-4}{5x-2}\right) = \frac{18}{(5x-2)^2}
\]
Solución
Let \( f(x) = \frac{x-4}{5x-2} \). We will use the Quotient Rule, which states that if
\[
f(x) = \frac{g(x)}{h(x)},
\]
then
\[
f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}.
\]
Here, let
\[
g(x) = x - 4 \quad \text{and} \quad h(x) = 5x - 2.
\]
First, calculate the derivatives of \( g(x) \) and \( h(x) \):
\[
g'(x) = 1,
\]
\[
h'(x) = 5.
\]
Now, apply the Quotient Rule:
\[
f'(x) = \frac{(5x-2)(1) - (x-4)(5)}{(5x-2)^2}.
\]
Simplify the numerator:
\[
(5x-2)(1) - (x-4)(5) = 5x - 2 - 5x + 20.
\]
Combine like terms:
\[
5x - 5x - 2 + 20 = 18.
\]
Thus, the derivative simplifies to:
\[
f'(x) = \frac{18}{(5x-2)^2}.
\]
So, the final result is:
\[
\frac{d}{dx}\left(\frac{x-4}{5x-2}\right) = \frac{18}{(5x-2)^2}.
\]
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Simplifique esta solución Mind Expander
To evaluate the derivative using the Quotient Rule, recall that if you have a function in the form \( \frac{u}{v} \), the derivative is given by \( \frac{u'v - uv'}{v^2} \). Here, let \( u = x - 4 \) and \( v = 5x - 2 \). First, compute \( u' \) and \( v' \): - \( u' = 1 \) - \( v' = 5 \) Now, apply the Quotient Rule: \[ \frac{d}{dx}\left(\frac{x-4}{5x-2}\right) = \frac{(1)(5x-2) - (x-4)(5)}{(5x-2)^2} \] Simplifying the numerator: \[ = \frac{(5x - 2) - (5x - 20)}{(5x - 2)^2} = \frac{5x - 2 - 5x + 20}{(5x - 2)^2} = \frac{18}{(5x - 2)^2} \] Thus, the simplified derivative is: \[ \frac{d}{dx}\left(\frac{x-4}{5x-2}\right) = \frac{18}{(5x-2)^2} \]
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