When a function z = f(x, y) depends on more than one independent variable (e.g., x and y), we can find its derivative with respect to one variable while treating all other variables as constants. This is called a **partial derivative**.
∂f/∂x = fₓ(x, y) = lim (h→0) [f(x + h, y) - f(x, y)] / h
∂f/∂y = fᵧ(x, y) = lim (k→0) [f(x, y + k) - f(x, y)] / k
If f(x, y) = 3x²y + 5y³ + sin(x)
∂f/∂x = 6xy + 0 + cos(x) = 6xy + cos(x) (y is constant)
∂f/∂y = 3x² + 15y² + 0 = 3x² + 15y² (x is constant)
We can take partial derivatives of the partial derivatives. These are called second-order partial derivatives.
∂²f/∂x² = fₓₓ = ∂/∂x (∂f/∂x)∂²f/∂y² = fᵧᵧ = ∂/∂y (∂f/∂y)∂²f/∂y∂x = fₓᵧ = ∂/∂y (∂f/∂x) (Differentiate w.r.t. x first, *then* w.r.t. y)∂²f/∂x∂y = fᵧₓ = ∂/∂x (∂f/∂y) (Differentiate w.r.t. y first, *then* w.r.t. x)Clairaut's Theorem (Symmetry of Mixed Partials): If fₓᵧ and fᵧₓ are both continuous, then they are equal: fₓᵧ = fᵧₓ. The order of differentiation does not matter.
A function f(x, y) is called **homogeneous of degree n** if replacing x with tx and y with ty results in the original function multiplied by tⁿ.
Definition: f(tx, ty) = tⁿ f(x, y)
Example: Let f(x, y) = x³ + 2x²y - y³
f(tx, ty) = (tx)³ + 2(tx)²(ty) - (ty)³
= t³x³ + 2(t²x²)(ty) - t³y³
= t³x³ + 2t³x²y - t³y³
= t³ (x³ + 2x²y - y³) = t³ f(x, y)
Therefore, the function is homogeneous of degree n = 3.
A homogeneous function can always be written in the form f(x, y) = xⁿ F(y/x) or f(x, y) = yⁿ G(x/y).
This theorem provides a beautiful relationship between a homogeneous function and its partial derivatives. The syllabus specifies the two-variable case.
Statement: Ifz = f(x, y)is a homogeneous function of degreen, then:
x(∂f/∂x) + y(∂f/∂y) = n·f
Example: For f(x, y) = x³ + 2x²y - y³ (degree n=3)
∂f/∂x = 3x² + 4xy
∂f/∂y = 2x² - 3y²
x(∂f/∂x) + y(∂f/∂y) = x(3x² + 4xy) + y(2x² - 3y²)
= 3x³ + 4x²y + 2x²y - 3y³
= 3x³ + 6x²y - 3y³
= 3 (x³ + 2x²y - y³) = 3·f
This verifies the theorem, as n = 3.
For a curve y = f(x) at a point P(x₁, y₁), the slope of the tangent line is given by the derivative m = dy/dx evaluated at that point.
Using the point-slope form Y - y₁ = m(X - x₁):
Equation of Tangent: Y - y₁ = (dy/dx)·(X - x₁)
The normal line is perpendicular to the tangent. Its slope is the negative reciprocal of the tangent's slope, m_normal = -1/m = -1/(dy/dx).
Equation of Normal: Y - y₁ = [-1 / (dy/dx)]·(X - x₁)
These are lengths measured along the x-axis.
|y₁ * sqrt(1 + (dx/dy)²)||y₁ * sqrt(1 + (dy/dx)²)|Length of Subtangent = |y₁ / (dy/dx)| = |y₁ · (dx/dy)|
Length of Subnormal = |y₁ · (dy/dx)|
For a polar curve r = f(θ), the geometry is different. We are often interested in the angle ϕ (phi) between the radius vector r and the tangent line.
The angle ϕ is given by the formula:
tan(ϕ) = r / (dr/dθ)
These are lengths on a line drawn *perpendicular* to the radius vector r through the pole (origin).
= |r² / (dr/dθ)|
= |dr/dθ|
f(x, y).f(tx, ty) and factoring out tⁿ to find the degree n.∂f/∂x and ∂f/∂y.x(∂f/∂x) + y(∂f/∂y) and show that the result simplifies to n·f.|y · (dy/dx)||dr/dθ| (This one is very simple!)tan(ϕ) = r / (dr/dθ). This formula is the key to all other polar tangent properties.