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Partial Derivatives

🔢 Partial Derivatives

Partial derivatives allow us to analyze how a function of multiple variables changes when we vary only one of those variables at a time.

🟢 1. Definition of Partial Derivative

For a function f(x,y)f(x, y), the partial derivative with respect to xx at (a,b)(a, b) is: fx(a,b)=∂f∂x=lim⁡h→0f(a+h,b)−f(a,b)hf_x(a, b) = \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(a+h, b) - f(a, b)}{h}

Similarly, the partial derivative with respect to yy is: fy(a,b)=∂f∂y=lim⁡h→0f(a,b+h)−f(a,b)hf_y(a, b) = \frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(a, b+h) - f(a, b)}{h}

How to Compute

To find ∂f∂x\frac{\partial f}{\partial x}, treat yy as a constant and differentiate with respect to xx using standard rules.

Example: If f(x,y)=x2y+sin⁡(x)f(x, y) = x^2 y + \sin(x), then:

  • ∂f∂x=2xy+cos⁡(x)\frac{\partial f}{\partial x} = 2xy + \cos(x)
  • ∂f∂y=x2\frac{\partial f}{\partial y} = x^2

🟡 2. The Multivariable Chain Rule

When the variables themselves depend on other variables, we use the multivariable Chain Rule.

Case 1: One Independent Variable

If z=f(x,y)z = f(x, y), where x=g(t)x = g(t) and y=h(t)y = h(t), then: dzdt=∂z∂xdxdt+∂z∂ydydt\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}

Case 2: Several Independent Variables

If z=f(x,y)z = f(x, y), where x=g(u,v)x = g(u, v) and y=h(u,v)y = h(u, v), then: ∂z∂u=∂z∂x∂x∂u+∂z∂y∂y∂u\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u} ∂z∂v=∂z∂x∂x∂v+∂z∂y∂y∂v\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v}

🔴 3. Higher-Order Partial Derivatives

We can take partial derivatives multiple times.

  • fxx=∂2f∂x2f_{xx} = \frac{\partial^2 f}{\partial x^2}
  • fyy=∂2f∂y2f_{yy} = \frac{\partial^2 f}{\partial y^2}
  • Mixed Partials: fxy=∂∂y(∂f∂x)f_{xy} = \frac{\partial}{\partial y} (\frac{\partial f}{\partial x}) and fyx=∂∂x(∂f∂y)f_{yx} = \frac{\partial}{\partial x} (\frac{\partial f}{\partial y}).

Clairaut’s Theorem

If ff and its partial derivatives fxyf_{xy} and fyxf_{yx} are continuous, then: fxy=fyxf_{xy} = f_{yx} The order of differentiation does not matter for mixed partials of “smooth” functions.

💡 Practical Example: Optimization (Tangent Planes)

The equation of the tangent plane to the surface z=f(x,y)z = f(x, y) at point (a,b,z0)(a, b, z_0) is: z−z0=fx(a,b)(x−a)+fy(a,b)(y−b)z - z_0 = f_x(a, b)(x - a) + f_y(a, b)(y - b)

def f(x, y):
    return x**2 + y**2

def df_dx(x, y):
    return 2*x

def df_dy(x, y):
    return 2*y

# At point (1, 2)
x0, y0 = 1, 2
z0 = f(x0, y0)
slope_x = df_dx(x0, y0)
slope_y = df_dy(x0, y0)

print(f"Tangent Plane: z - {z0} = {slope_x}(x - {x0}) + {slope_y}(y - {y0})")

🚀 Key Takeaways

  • Partial derivatives measure change along coordinate axes.
  • The multivariable Chain Rule sums up contributions from each path.
  • Mixed partials are equal for most smooth functions.