Mathematics

Problem

The curve defined implicitly by

x^{2}y+\ln y=x

passes through the point

(1,1)

. (a) Derive an explicit expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Differentiate again to obtain

\dfrac{d^{2}y}{dx^{2}}

in terms of

x

and

y

, simplifying your result. (c) Compute

\dfrac{d^{3}y}{dx^{3}}

at the point

(1,1)

. Present a clear sequence of differentiations leading to your answer. (d) Determine whether

(1,1)

is a point of inflection for the curve. Provide a rigorous justification that references the second derivative and the value obtained for the third derivative.

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