Question

Find dx/dy. Related Rates. Help me please.

Answer 1

`\frac{\text{d}x}{\text{d}t}= -(100)/(3)`

Explanation:

To find dx/dt when x = 5, first differentiate xy = 3 with respect to x using implicit differentiation.

Begin by placing d/dx in front of each term of the equation:

`\frac{\text{d}}{\text{d}x}xy=\frac{\text{d}}{\text{d}x}3`

Differentiate the constant only:

`\frac{\text{d}}{\text{d}x}xy=0`

Use the product rule to differentiate xy with respect to x:

`\textsf{Let}\;u=x \implies \frac{\text{d}u}{\text{d}x}=1`

`\textsf{Let}\;v=y \implies \frac{\text{d}v}{\text{d}x}=1\frac{\text{d}y}{\text{d}x}=\frac{\text{d}y}{\text{d}x}`

Substitute u, v, du/dx and dv/dx into the chain rule formula:

`\frac{\text{d}}{\text{d}x}xy=u\cdot \frac{\text{d}v}{\text{d}x}+v \cdot \frac{\text{d}u}{\text{d}x}`

`\frac{\text{d}}{\text{d}x}xy=x\cdot \frac{\text{d}y}{\text{d}x}+y \cdot 1`

`\frac{\text{d}}{\text{d}x}xy=x\frac{\text{d}y}{\text{d}x}+y`

Therefore, xy = 3 differentiated with respect to x is:

`x\frac{\text{d}y}{\text{d}x}+y=0`

Rearrange to make dy/dx the subject:

`\frac{\text{d}y}{\text{d}x}=-(y)/(x)`

To find the equation for dx/dt, we can use the chain rule:

`\frac{\text{d}x}{\text{d}t}=\frac{\text{d}x}{\text{d}y}* \frac{\text{d}y}{\text{d}t}`

Substitute dx/dy and dy/dt = 4 into the equation:

`\frac{\text{d}x}{\text{d}t}=-(x)/(y)*4`

`\frac{\text{d}x}{\text{d}t}=-(4x)/(y)`

Find the corresponding value of y when x = 5 by substituting x = 5 into the original equation:

`\begin{aligned}x=5 \implies 5y&=3\\y&=(3)/(5)\end{aligned}`

Finally, to find dx/dt when x = 5, substitute x = 5 and y = 3/5 into dx/dt:

`\frac{\text{d}x}{\text{d}t}=-(4(5))/((3)/(5))`

`\frac{\text{d}x}{\text{d}t}=-(20)/((3)/(5))`

`\frac{\text{d}x}{\text{d}t}=-20\cdot (5)/(3)`

`\frac{\text{d}x}{\text{d}t}= -(100)/(3)`

Therefore, the value of dx/dt when x = 5 is -100/3.