If x=y (x y), then \\;d y/d x is equal to:

Correct Answer is : ;x−yx(1+log⁡xy)\\;x-y/x(1+ x y);x(1+logxy)x−y

Correct Answer is : ;x−yx(1+log⁡xy)\\;x-y/x(1+ x y);x(1+logxy)x−y

Test Index

Differential Calculus

Question : 26 of 69

Marks: +1, -0

x=y \log (x y),

\\;\frac{d y}{d x}

is equal to:

Solution:

Concept:

Implicit differentiation using product and chain rules.

Explanation:

Given

x = y \log(xy)

. Differentiate both sides with respect to

x

Left side:

\frac{d}{dx}(x) = 1

Right side: Use product rule:

\frac{d}{dx}[y \log(xy)] = \frac{dy}{dx} \log(xy) + y \cdot \frac{d}{dx}[\log(xy)]

Now

\frac{d}{dx}[\log(xy)] = \frac{1}{xy} \cdot \frac{d}{dx}(xy) = \frac{1}{xy}(y + x\frac{dy}{dx}) = \frac{y + x\frac{dy}{dx}}{xy}

Thus the derivative becomes

\frac{dy}{dx} \log(xy) + y \cdot \frac{y + x\frac{dy}{dx}}{xy} = \frac{dy}{dx} \log(xy) + \frac{y^2 + xy\frac{dy}{dx}}{xy} = \frac{dy}{dx} \log(xy) + \frac{y}{x} + \frac{dy}{dx}

Set equal to 1:

1 = \frac{dy}{dx}(\log(xy) + 1) + \frac{y}{x}

Rearrange:

\frac{dy}{dx}(\log(xy) + 1) = 1 - \frac{y}{x} = \frac{x - y}{x}

Hence

\frac{dy}{dx} = \frac{x - y}{x(1 + \log(xy))}

Answer:

\frac{dy}{dx} = \frac{x - y}{x(1 + \log(xy))}

, which corresponds to option B.

Go to Question:

Prev Question

Next Question