Partial Derivatives

Concept Explanation

In multivariable space, a surface has different slopes depending on the direction of travel. Partial derivatives measure the rate of change directly along the principal coordinate axes. Combining these partial derivatives yields the Gradient vector ($\nabla f$), pointing in the direction of steepest ascent.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Engineers use partial derivatives to model heat diffusion gradients across metal plates, solve electrostatics potential fields, and optimize multi-variable parameters like fluid pressure distribution.

Example Problem

Compute the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for:

Solution

Step 1: Compute $\frac{\partial f}{\partial x}$ (treat $y$ as a constant)

$$\frac{\partial f}{\partial x} = \frac{d}{dx}(x^2) \cdot y^3 + \frac{d}{dx}(5x)$$

$$= 2xy^3 + 5$$

Step 2: Compute $\frac{\partial f}{\partial y}$ (treat $x$ as a constant)

$$\frac{\partial f}{\partial y} = x^2 \cdot \frac{d}{dy}(y^3) + \frac{d}{dy}(5x)$$

$$= 3x^2 y^2 + 0 = 3x^2 y^2$$

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Connections & References

• Parent Concepts: Derivatives, Vector Valued Functions
• Sub-concepts: Multiple Integrals
• Course Links: Calc-3 Session 09 | Calc-3 Session 12