First slide

Derivatives

Question

$If (a + b x) e^{\frac{y}{x}} = x then x^{3} \frac{d^{2} y}{d x^{2}} =$

Moderate

A

${(x \frac{d y}{d x} - y)}^{2}$
B

${(\frac{d y}{d x})}^{2}$
C

0
D

1

Solution

$(a + b x) e^{y / x} = x \Rightarrow e^{y / x} = \frac{x}{a + b x} \Rightarrow \frac{y}{x} = \log (\frac{x}{a + b x}) = \log x - \log (a + b x)$
$\frac{d}{d x} (\frac{y}{x}) = \frac{d}{d x} [\log x - \log (a + b x)] \Rightarrow \frac{x \frac{d y}{d x} - y 1}{x^{2}} = \frac{1}{x} - \frac{b}{a + b x} = \frac{a}{x (a + b x)} \Rightarrow x \frac{d y}{d x} - y = \frac{a x}{a + b x} \to (1)$

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