Expert Answer to "How to find the derivatives of x =..."

Gianna Johnson

Answered question

2023-03-21

How to find the derivatives of

x = \ln (x y)

pyroleo7zjv

Beginner2023-03-22Added 5 answers

Take the exponential of both sides of the equation:

x = \ln (x y) \Rightarrow e^{x} = e^{\ln (x y)} = x y

Thus, y(x) can be made explicit:

y (x) = \frac{e^{x}}{x}

and

\frac{d y}{d x} = \frac{x e^{x} - e^{x}}{x^{2}} = e^{x} (\frac{x - 1}{x^{2}})

Ralph Sparks

Beginner2023-03-23Added 8 answers

Use the properties of logarithms and its inverse to write the given equation as a function of y and then use the quotient rule:
Explanation: Given:

x = \ln (x y)

x = \ln (x) + \ln (y)

\ln (y) = x - \ln (x)

\ln (y) = x + \ln (\frac{1}{x})

e^{\ln (y)} = e^{x + \ln (\frac{1}{x})}

y = \frac{e^{x}}{x}

\frac{d y}{d x} = \frac{x - 1}{x^{2}} e^{x}

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