Calculus guide
Gradients and Multivariable Calculus
Compute partial derivatives, assemble the gradient vector, and interpret the direction of fastest increase.
9 min readIntermediateUpdated 2026-07-12
What you will learn
- Hold other variables constant in a partial derivative
- Build a gradient vector
- Interpret gradient direction and magnitude
Concept 1
Partial derivatives change one variable at a time
When differentiating with respect to x, treat y and z as constants. Repeat for each independent variable.
Concept 2
The gradient collects all first partials
The gradient is a vector field. At a point where it is nonzero, it points in the direction of greatest local increase.
Concept 3
Directional derivatives use a unit direction
Project the gradient onto a unit vector u. The largest possible directional derivative is the gradient magnitude.
Worked example
Find ∇f for f(x,y)=x²+y²
- 1Differentiate with respect to x while holding y constant: fₓ=2x.
- 2Differentiate with respect to y while holding x constant: fᵧ=2y.
- 3Place the partial derivatives into a vector.
Answer
Common mistakes
- Differentiating every variable at once
- Writing a scalar when a gradient vector is required
- Using a direction vector without normalizing it
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