Cue Column:

• When is implicit differentiation needed?
• How to handle $y^n$ terms?
• How to find $d^{2y}$/$dx^2$ implicitly?

Notes Column: Implicit differentiation is used when y is not explicitly expressed as a function of x, i.e., when we have F(x, y) = 0.

Method:

1. Differentiate both sides with respect to x
2. When differentiating a term involving y, apply chain rule: d/dx(f(y)) = f'(y) * dy/dx
3. Collect all terms with dy/dx on one side
4. Solve for dy/dx

Example: $x^2$ + $y^2$ = $a^2$

• 2x + 2y * $\frac{dy}{dx}$ = 0
• dy/dx = -x/y

Example: $x^3$ + $y^3$ = 3axy (Folium of Descartes)

• 3$x^2$ + 3$y^2$ * $\frac{dy}{dx}$ = 3a[y + x * $\frac{dy}{dx}$]
• 3$y^2$ * $\frac{dy}{dx}$ - 3ax * $\frac{dy}{dx}$ = 3ay - 3$x^2$
• dy/dx = $\frac{ay - x^2}{(y^2 - ax)}$

For second derivative: differentiate dy/dx again with respect to x, substituting the first derivative expression where needed.

Summary: Differentiate everything with respect to x, remember that y is a function of x (so d/dx of y terms gets a dy/dx factor), then solve algebraically for dy/dx.