`4xy + ln (x^2 y) = 7` Use implicit differentiation to find dy/dx

Find $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ if $\displaystyle{4}{x}{y}+{\ln{{\left({{x}}^{{2}}{y}\right)}}}={7}$

Use properties of logarithms to rewrite the second term:

$\displaystyle{4}{x}{y}+{2}{\ln{{x}}}+{\ln{{y}}}={7}$

Differentiate term by term with respect to x:

$\displaystyle{4}{y}+{4}{x}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}+\frac{{2}}{{x}}+\frac{{1}}{{y}}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={0}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}{\left({4}{x}+\frac{{1}}{{y}}\right)}=-{\left(\frac{{2}}{{x}}+{4}{y}\right)}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=-\frac{{\frac{{2}}{{x}}+{4}{y}}}{{{4}{x}+\frac{{1}}{{y}}}}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=-\frac{{{2}{y}+{4}{x}{{y}}^{{2}}}}{{{4}{{x}}^{{2}}{y}+{x}}}$

Ninenotes

Friday, May 22, 2015

$\displaystyle{4}{x}{y}+{\ln{{\left({{x}}^{{2}}{y}\right)}}}={7}$ Use implicit differentiation to find dy/dx

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