Sometimes y is defined implicitly (not solved for y), e.g. \(x^2 + y^2 = 25\) (circle).
Differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\):
\(2x + 2y dy/dx = 0\)
\(dy/dx = -x/y\) (slope of tangent at any point on circle)
Very useful when \(y\) cannot be easily isolated.
When quantities change over time, their rates are related.
Example: circle radius \(r\) increases at \(dr/dt = 2\) cm/s. How fast is area increasing when r = 5 cm?
\(A = \pi r^2\)
\(dA/dt = 2\pi r dr/dt\)
When \(r = 5,\, dA/dt = 2\pi·5·2 = 20\pi\) cm²/s
Steps: relate variables → differentiate both sides w.r.t. time → plug in known rates.