Question
Find \( \frac{d y}{d x} \) by implicit differentiation. Then find the slope of the graph at the given polnt, (IF an answer is undefined, enter unDEFINED.) \( \frac{d y}{d x}=x y=30,(-6,-5) \) At \( (-6,-5) ; y^{\prime}= \)
Ask by Lawson Dawson. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
At (-6, -5), the slope \( y' = -\frac{5}{6} \).
Solution
We start with the equation
x · y = 30.
Differentiate both sides with respect to x using the product rule on the left side:
d/dx (x · y) = (d/dx x) · y + x · (d/dx y) = y + x · (dy/dx).
Since 30 is a constant, its derivative is 0:
y + x · (dy/dx) = 0.
Now, solve for dy/dx:
x · (dy/dx) = -y ⟹ dy/dx = -y/x.
To find the slope at the point (-6, -5), substitute x = -6 and y = -5:
dy/dx = -(-5)/(-6) = (5)/(-6) = -5/6.
Thus, the slope of the graph at (-6, -5) is -5/6.
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Simplify this solution Mind Expander
To find \( \frac{d y}{d x} \) by implicit differentiation from the equation \( xy = 30 \), we start by differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(xy) = \frac{d}{dx}(30) \] Applying the product rule on the left side: \[ x \frac{dy}{dx} + y \frac{d}{dx}(x) = 0 \] This simplifies to: \[ x \frac{dy}{dx} + y = 0 \] Now, solve for \( \frac{dy}{dx} \): \[ x \frac{dy}{dx} = -y \] \[ \frac{dy}{dx} = -\frac{y}{x} \] Now, we can substitute the given point \((-6, -5)\) into the expression for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = -\frac{-5}{-6} = \frac{5}{6} \] Thus, at the point \((-6, -5)\): \[ y' = \frac{5}{6} \]
