Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

xy=2

(a) Find `dy/dt` ,given `x=2` and `dx/dt =11`

(b) Find `dx/dt` , given `x=1` and `dy/dt=-7`

If someone can explain the steps so I understand these types of problems it would mean a lot.

Thanks

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The given equation is:

`xy=2`

Take the derivative with respect to t on both sides of the equation.

`d/dt(xy)=d/dt(2)`

Note that the derivative of constant is zero. So right side becomes:

`d/dt(xy)=0`

For the right side, apply the power rule which is`(u*v)'=v*u' + u*v'`.

`ydx/dt+xdy/dt=0`

Now that derivative of the equation with respect to x is known, use this and the given equation to solve the two problems.

(A) `x=2` and `dx/dt=11` , `dy/dt=?`

Before substituting the given values to the derivative, solve for y first.

To do so, plug-in x=2 to:

`xy=2`

`2y=2`

`y=1`

Now that value of y is known, plug-in x=2, y=1 and `dx/dt=11` to:

`ydx/dt+xdy/dt=0`

`1(11)+2(dy)/(dt)=0`

`11+2dy/dt=0`

`2dy/dt=-11`

`dy/dt=-11/2`

**Hence, when `x=2` and `dx/dt=11` , `dy/dt=-11/2` .**

(B) `x=1` and `dy/dt=-7` , `dx/dt= ?`

Do the same steps as above. Plug-in x=1 to the given equation to solve for y.

`xy=2`

`1*y=2`

`y=2`

Then, substitute x=1, y=2 and `dy/dt=-7` to:

`ydx/dt+xdy/dt=0`

`2dx/dt+1*(-7)=0`

`2dx/dt-7=0`

`2dx/dt=7`

`dx/dt=7/2`

**Hence, when `x=1` and `dy/dt=-7` , `dx/dt=7/2` .**

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