Question

How do you use implicit differentiation to find the slope of the curve given `12(x^2+y^2)=25xy` at (3,4)?

Answer 1

Please see the explanation.

Explanation:

Given: `12(x^2+y^2)=25xy`

Use the distributive property:

`12x^2+12y^2=25xy`

Differentiate each term:

`d/dx12x^2+d/dx12y^2=d/dx25xy`

The first term is just the power rule:

`24x+d/dx12y^2=d/dx25xy`

The second term requires the chain rule:

`24x+24ydy/dx=d/dx25xy`

The last term requires the use of the product rule:

`24x+24ydy/dx=25y+ 25xdy/dx`

Collect all of the terms containing `dy/dx` on the left and the other terms on the right:

`24ydy/dx-25xdy/dx=25y-24x`

Factor out `dy/dx`:

`(24y-25x)dy/dx=25y-24x`

Divide both sides by the leading factor on the left:

`dy/dx=(25y-24x)/(24y-25x)`

The slope, m, of the tangent line is the above evaluated at `(3,4)`:

`m = (25(4)-24(3))/(24(4)-25(3))`

`m = 4/3`