Question

50 points each question. Please help. How do I solve?

Answer 1

`~~~~~\ln(xy) +4x = 45\\\\\\\implies (d)/(dx) \left[\ln(xy) +4x \right] = (d)/(dx) ( 45)\\\\\\\implies (1)/(xy) \cdot (d)/(dx) (xy) + 4 = 0\\\\\\\implies \frac 1{xy} \left( x(dy)/(dx) + y \right) +4 =0\\\\\\\implies x(dy)/(dx) + y + 4xy=0\\\\\\\implies x (dy)/(dx) =-y-4xy\\\\\\\implies (dy)/(dx) = \frac{-y-4xy}x`

Answer 2

`(dy)/(dx)=-y\left(4+(1)/(x)\right)`

or written in rational form:

`(dy)/(dx)=(-4xy-y)/(x)`

Explanation:

`\ln(xy)+4x=45`

`\textsf{Apply the product log law}\quad\ln(ab)=\ln(a)+\ln(b):`

`\implies \ln(x)+\ln(y)+4x=45`

`\textsf{Add}\:(d)/(dx)\:\textsf{in front of each term}:`

`\implies (d)/(dx)\ln(x)+(d)/(dx)\ln(y)+(d)/(dx)4x=(d)/(dx)45`

`\textsf{Differentiate the}\: x \: \textsf{terms and constant terms first}:`

`\implies (1)/(x)+(d)/(dx)\ln(y)+4=0`

`\textsf{When differentiating with respect to}\:y,\: \textsf{differentiate and add}\: (dy)/(dx):`

`\implies (1)/(x)+(1)/(y)(dy)/(dx)+4=0`

`\textsf{Rearrange to make}\:(dy)/(dx)\:\textsf{the subject}:`

`\implies (1)/(y)(dy)/(dx)=-4-(1)/(x)`

`\implies (dy)/(dx)=y\left(-4-(1)/(x)\right)`

`\implies (dy)/(dx)=-y\left(4+(1)/(x)\right)`

If you want it in rational form, then:

`\implies (dy)/(dx)=-4y-(y)/(x)`

`\implies (dy)/(dx)=(-4xy-y)/(x)`