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18. [-/1 Points] DETAILS MY NOTE Find \( \frac{d y}{d x} \) by implicit differentiation. \( e^{x / y}=5 x-y \) \( \frac{d y}{d x}=\square \)

Ask by Malone West. in the United States
Mar 18,2025

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\[ \frac{dy}{dx} = \frac{e^{x/y} \cdot y - 5y^2}{e^{x/y} \cdot x - y^2} \]

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Let’s have some fun with implicit differentiation! Start with the equation \( e^{x/y} = 5x - y \). Differentiate both sides with respect to \( x \). For the left side, you’ll need to use the chain rule, which brings in \( \frac{d y}{d x} \) when you differentiate \( y \). So once you differentiate, you'll have: \[ \frac{d}{dx}\left(e^{x/y}\right) = e^{x/y} \left(\frac{y - x\frac{d y}{d x}}{y^2}\right) \] On the right side, the derivative of \( 5x \) is \( 5 \), and for \( -y \), it's \( -\frac{d y}{d x} \). Set them equal to each other and solve for \( \frac{d y}{d x} \). Now, after a bit of algebra (and some cleverness), you’ll find: \[ \frac{d y}{d x} = \frac{5y}{e^{x/y} \cdot (y - x\frac{d y}{d x}) + 1} \] And voilà! You’ve got \( \frac{d y}{d x} \) ready to tackle further problems.

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