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Question
If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ______.
Options
2
0
–1
1
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Solution
If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = 1.
Explanation:
log(x + y) = 2xy ...(1)
∴ `(1)/"x + y".(1 + "dy"/"dx") = 2x"dy"/"dx" + 2y`
∴ `(1/(x + y) - 2x)"dy"/"dx" = 2y - (1)/"x + y"`
∴ `"dy"/"dx" = (2y(x + y) - 1)/(1 - 2x(x + y)`
If x = 0, then from (1),
log y = 0 = log 1
∴ y = 1
∴ y'(0) = `(2(1)(0 + 1) - 1)/(1 - 2(0)(0 + 1))` = 1.
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