Question
Find the derivative of the function. \[ y=e^{-4 / x} \]
Ask by Powell Coles. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( y = e^{-4/x} \) is \( \frac{dy}{dx} = \frac{4e^{-4/x}}{x^2} \).
Solution
1. Let \( u = -\frac{4}{x} \) so that \( y = e^u \).
2. Differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = \frac{d}{dx}\left(-\frac{4}{x}\right) = -4 \cdot \frac{d}{dx}\left(\frac{1}{x}\right) = -4 \cdot \left(-\frac{1}{x^2}\right) = \frac{4}{x^2}.
\]
3. Apply the chain rule for differentiation, which gives:
\[
\frac{dy}{dx} = e^u \cdot \frac{du}{dx}.
\]
4. Substitute back for \( u \):
\[
\frac{dy}{dx} = e^{-4/x} \cdot \frac{4}{x^2}.
\]
The derivative of the function is:
\[
\frac{dy}{dx} = \frac{4e^{-4/x}}{x^2}.
\]
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Simplify this solution Extra Insights
To find the derivative of the function \( y = e^{-4/x} \), we can use the chain rule. The derivative of \( e^{u} \) is \( e^{u} \cdot \frac{du}{dx} \), where \( u = -\frac{4}{x} \). First, we calculate \( \frac{du}{dx} \): \[ u = -\frac{4}{x} = -4x^{-1} \] Using the power rule: \[ \frac{du}{dx} = -4 \cdot (-1)x^{-2} = \frac{4}{x^2} \] Now, applying the chain rule: \[ \frac{dy}{dx} = e^{-4/x} \cdot \frac{du}{dx} = e^{-4/x} \cdot \frac{4}{x^2} \] Thus, the derivative of the function is: \[ \frac{dy}{dx} = \frac{4e^{-4/x}}{x^2} \]
