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Question
Find `dy/dx` for the function given in the question:
`xy = e^((x – y))`
Solution
Given, xy = e(x-y)
Taking logarithm of both the sides,
log (xy) = log e(x-y)
or log x + log y = (x - y) loge e .... [∵ log xy = log x + log y]
or log x + log y = x - y ...[∵ loge e = 1]
Differentiating both sides with respect to x,
`d/dx log x +d/dx log y = d/dx (x) - d/dx (y)`
or `1/x + 1/y dy/dx = 1 - dy/dx `
or`1/y dy/dx + dy/dx = 1 - 1/x`
or `dy/dx ((1 + y)/y) = 1 - 1/x = (x - 1)/x`
or `((1 + y)/y) dy/dx = (x - 1)/x`
`therefore dy/dx = (y (x - 1))/(x (1 + y))`
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