Pregunta
Find the slope of the following curve at \( x=8 \) \( y=\frac{1}{x-4} \) The slope of the given curve at \( x=8 \) is (Simplify your answer.)
Ask by Davison Potter. in the United States
Mar 09,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The slope of the curve at \( x=8 \) is \( -\frac{1}{16} \).
Solución
1. Given the function
\[
y = \frac{1}{x-4},
\]
we rewrite it as
\[
y = (x-4)^{-1}.
\]
2. Differentiate \( y \) with respect to \( x \) using the power rule:
\[
\frac{dy}{dx} = -1 \cdot (x-4)^{-2} \cdot \frac{d}{dx}(x-4).
\]
Since \(\frac{d}{dx}(x-4)=1\), we have
\[
\frac{dy}{dx} = -\frac{1}{(x-4)^2}.
\]
3. Evaluate the derivative at \( x=8 \):
\[
\frac{dy}{dx}\Big|_{x=8} = -\frac{1}{(8-4)^2} = -\frac{1}{4^2} = -\frac{1}{16}.
\]
The slope of the given curve at \( x=8 \) is
\[
-\frac{1}{16}.
\]
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Como
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Mind Expander
To find the slope of the curve \( y = \frac{1}{x-4} \) at \( x = 8 \), we first need to compute the derivative, \( y' \). Using the quotient rule, we have: \[ y' = -\frac{1}{(x-4)^2} \] Now, we substitute \( x = 8 \) into the derivative: \[ y' = -\frac{1}{(8-4)^2} = -\frac{1}{4^2} = -\frac{1}{16} \] Thus, the slope of the given curve at \( x=8 \) is \( -\frac{1}{16} \).

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