How to differentiate x^2 + y^2 - 2x + 6y = 5

This is an implicit function, meaning the x and y terms are not able to be seperated onto different sides, i.e. y=f(x). Therefore, we must use implicit differentiation

Before we differentiate the equation given, recall that when differentiating y with respect to (w.r.t) x, we get an answer of dy/dx. This is because we have differentiated y normally w.r.t y first, and then multiplied it by dy/dx.  For example, if we differentiate 2y w.r.t x, we get 2 dy/dx

Now, in order to differentiate the equation above, take it one term at a time, as follows:

1. Differentiate x2 w.r.t x, which gives 2x

2. Differentiate y2: differentiate y2 w.r.t y first, which gives 2y, then multiply by dy/dx, giving 2y dy/dx.

3. Differentiate -2x => -2

4. Differentiate 6y => 6 dy/dx 

5. (And on the right side) differentiate 5 => 0

Then, put all the terms together, in the new differentiated equation:

2x + 2y dy/dx - 2 + 6 dy/dx = 0 

The final step is simply to rearrange the above equation to make dy/dx the subject of the formula. When done you should get:

dy/dx = (2 - 2x) / (2y + 6)

And there you have it! 

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