Question

How do you find `(dy)/(dx)` given `sqrty+xy^2=5`?

Answer 1

`color(blue)(-(2y^(5/2))/(1+4xy^(3/2)))`

Explanation:

We need to differentiate this implicitly, because we don't have a function in terms of one variable.

When we differentiate `y` we use the chain rule:

`d/dy*dy/dx=d/dx`

As an example if we had:

`y^2`

This would be:

`d/dy(y^2)*dy/dx=2ydy/dx`

In this example we also need to use the product rule on the term `xy^2`

Writing `sqrt(y)` as `y^(1/2)`

`y^(1/2)+xy^2=5`

Differentiating:

`1/2y^(-1/2)*dy/dx+x*2ydy/dx+y^2=0`

`1/2y^(-1/2)*dy/dx+x*2ydy/dx=-y^2`

Factor out `dy/dx`:

`dy/dx(1/2y^(-1/2)+2xy)=-y^2`

Divide by `(1/2y^(-1/2)+2xy)`

`dy/dx=(-y^2)/((1/2y^(-1/2)+2xy))=(-y^2)/(1/(2sqrt(y))+2xy`

Simplify:

Multiply by: `2sqrt(y)`

`(-y^2*2sqrt(y))/(2sqrt(y)1/(2sqrt(y))+2xy*2sqrt(y)`

`(-y^2*2sqrt(y))/(cancel(2sqrt(y))1/(cancel(2sqrt(y)))+2xy*2sqrt(y)`

`(-y^2*2sqrt(y))/(1+2xy*2sqrt(y))=-(2sqrt(y^5))/(1+4xsqrt(y^3))=color(blue)(-(2y^(5/2))/(1+4xy^(3/2)))`