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:
- Differentiate both sides with respect to x
- When differentiating a term involving y, apply chain rule: d/dx(f(y)) = f'(y) * dy/dx
- Collect all terms with dy/dx on one side
- Solve for dy/dx
Example: x^2 + y^2 = a^2
- 2x + 2y * (dy/dx) = 0
- dy/dx = -x/y
Example: x^3 + y^3 = 3axy (Folium of Descartes)
- 3x^2 + 3y^2 * (dy/dx) = 3a[y + x * (dy/dx)]
- 3y^2 * (dy/dx) - 3ax * (dy/dx) = 3ay - 3x^2
- dy/dx = (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.