When: the equation mixes $x$ and $y$ and is hard to solve for $y$ , like $x^2+y^2=25$ or $x^3+y^3=6xy$ .

Recipe:

Differentiate both sides with respect to $x$ . Every $y$ -term gets a $y'$ (chain rule).
Move all $y'$ terms to one side, everything else to the other.
Factor out $y'$ and divide.

Building blocks:

$(y^n)'=n\,y^{n-1}\,y'$
$(xy)'=y+x\,y'$ (product rule)
$(x^2y)'=2xy+x^2y'$
$(\sin y)'=\cos y\cdot y'$ , $(e^y)'=e^y\,y'$
A constant on either side → 0.

Example. $x^2+y^2=25$ : $2x+2yy'=0\Rightarrow y'=-\dfrac{x}{y}$ . At $(3,4)$ the slope is $-\tfrac34$ .

Slope at a point, fast: plug in $x$ and $y$ right after differentiating, then solve for $y'$ . For $x^2+xy+y^2=7$ at $(1,2)$ : $2+2+y'+4y'=0\Rightarrow y'=-\tfrac45$ .

Horizontal tangent: numerator of $y'$ is 0. Vertical tangent: denominator is 0.

Definition (2.8)

Basic rules (3.1)

Product / Quotient (3.2)

Trig (3.3)

Chain (3.4)

Implicit (3.5)