Implicit and Parametric Differentiation
Implicit and Parametric Differentiation
Not all curves can be written as y = f(x). Implicit differentiation handles equations where y is tangled with x, while parametric differentiation handles curves defined through a third variable (the parameter).
Implicit Differentiation
An implicitly defined curve is one where x and y are mixed together in one equation, such as x² + y² = 25 (a circle). To find dy/dx:
1. Differentiate every term with respect to x
2. When differentiating a term involving y, apply the chain rule: d/dx[f(y)] = f'(y) × dy/dx
3. Collect all dy/dx terms on one side and solve for dy/dx
Worked Example 1: Find dy/dx for x² + y² = 25.
Differentiate each term with respect to x:
2x + 2y·(dy/dx) = 0
2y·(dy/dx) = -2x
dy/dx = -x/y
This gives the gradient at any point (x, y) on the circle. At (3, 4): dy/dx = -3/4.
Worked Example 2: Find dy/dx for x³ + 3xy + y³ = 8.
Differentiate term by term. The term 3xy requires the product rule:
d/dx(3xy) = 3[x·(dy/dx) + y·1] = 3x(dy/dx) + 3y
So the full differentiation gives:
3x² + 3x(dy/dx) + 3y + 3y²(dy/dx) = 0
Collect dy/dx terms: (3x + 3y²)(dy/dx) = -3x² - 3y
dy/dx = -(x² + y) / (x + y²)
Tangents and Normals from Implicit Equations
Once you have dy/dx in terms of x and y, substitute the coordinates of the point to find the gradient. Then use y - y₁ = m(x - x₁) for the tangent, and the negative reciprocal gradient for the normal.
Worked Example: Find the equation of the tangent to x² + xy + y² = 7 at (1, 2).
Differentiate: 2x + x(dy/dx) + y + 2y(dy/dx) = 0
(x + 2y)(dy/dx) = -2x - y
dy/dx = -(2x + y)/(x + 2y)
At (1, 2): dy/dx = -(2 + 2)/(1 + 4) = -4/5
Tangent: y - 2 = -4/5(x - 1), which gives 4x + 5y = 14.
Parametric Equations
A curve can be defined by expressing x and y separately in terms of a parameter (usually t or θ):
x = f(t), y = g(t)
Common examples include:
- Circle: x = rcosθ, y = rsinθ
- Parabola: x = at², y = 2at
- Ellipse: x = acosθ, y = bsinθ
Parametric Differentiation
To find dy/dx for parametric curves, use the chain rule:
dy/dx = (dy/dt) / (dx/dt)
This is valid provided dx/dt ≠ 0.
Worked Example 1: A curve is defined by x = t² + 1, y = t³ - t. Find dy/dx.
dx/dt = 2t and dy/dt = 3t² - 1
dy/dx = (3t² - 1) / (2t)
At t = 2: dy/dx = (12 - 1)/4 = 11/4
Worked Example 2: A curve is defined by x = 3cosθ, y = 3sinθ. Find dy/dx.
dx/dθ = -3sinθ and dy/dθ = 3cosθ
dy/dx = 3cosθ / (-3sinθ) = -cosθ/sinθ = -cotθ
Second Derivatives with Parameters
To find d²y/dx², differentiate dy/dx with respect to t, then divide by dx/dt:
d²y/dx² = (d/dt(dy/dx)) / (dx/dt)
Worked Example: For x = t², y = t³, find d²y/dx².
dy/dx = 3t²/2t = 3t/2
d/dt(dy/dx) = d/dt(3t/2) = 3/2
d²y/dx² = (3/2) / (2t) = 3/(4t)
Converting Between Parametric and Cartesian Forms
To convert parametric equations to Cartesian form, eliminate the parameter:
- Rearrange one equation for t
- Substitute into the other
- For trig parameters, use sin²θ + cos²θ = 1
Example: x = 2t, y = t² becomes t = x/2, so y = (x/2)² = x²/4.
Example: x = 5cosθ, y = 5sinθ. Then x² + y² = 25cos²θ + 25sin²θ = 25.
Exam Tips
- In implicit differentiation, never forget dy/dx when differentiating y terms — this is the most common error.
- Products of x and y (like 3xy) always need the product rule.
- For parametric tangent/normal questions, find the value of the parameter at the given point first, then compute dy/dx.
- When finding where a parametric curve crosses the x-axis, set y = 0 (not t = 0).
- Check whether a question asks for the answer in terms of the parameter or in Cartesian form.