Partial Derivatives, Gradients, and the Total Derivative
Core ideas about differentiating multivariable functions as linear maps, plus gradients and directional derivatives.
Keyboard Shortcuts
💡 Pro tip: Use keyboard shortcuts for faster studying!
Study Smart Tips for Partial Derivatives, Gradients, and the Total Derivative
Master these concepts using proven study techniques that actually work:
Active Recall
Test yourself before flipping each card to strengthen memory retention
Spaced Repetition
Review difficult cards more frequently than easy ones
Multiple Sessions
Break study time into shorter, focused sessions
Explain Aloud
Verbalize answers to reinforce understanding
Questions Covered in This Set
10 cards to master
What is the one big idea about the derivative of a multivariable map?
The derivative is not a number but a linear map (written as a matrix) that gives the best linear approximation to the function near a point.
How do you compute ∂f/∂x operationally?
Differentiate with respect to x while treating all other variables as constants; geometrically it's the slope of the slice y = b of the surface z = f(x,y).
Compute f_x and f_y for f(x,y) = x³y² + sin(xy) + 5y.
f_x = 3x²y² + y·cos(xy); f_y = 2x³y + x·cos(xy) + 5.
What does Clairaut's theorem say?
If the second partials are continuous, mixed partials commute: f_xy = f_yx. This underwrites exact ODEs, conservative vector fields, and symmetry of the Hessian.
Define the gradient ∇f.
For f : ℝⁿ → ℝ, ∇f = (∂f/∂x₁, …, ∂f/∂xₙ), the vector of all first partial derivatives.
Formula for the directional derivative in a unit direction u.
D_u f(a) = lim_{h→0} [f(a+hu) − f(a)]/h = ∇f(a) · u = ‖∇f‖cos θ.
Three geometric consequences of D_u f = ‖∇f‖cos θ.
(1) f increases fastest along ∇f at rate ‖∇f‖; (2) decreases fastest along −∇f (gradient descent); (3) ∇f is perpendicular to level sets.
For f(x,y) = x² + 3y² at (1,2), find ∇f and the max rate of increase.
∇f = (2x, 6y) = (2,12); steepest ascent direction (2,12)/√148 with rate √148 ≈ 12.17.
How do you get a tangent plane to the surface g(x,y,z) = c?
Use the gradient as normal vector: ∇g(a) · (x − a) = 0.
Does having all partial derivatives imply differentiability?
No. Existence of partials is not sufficient; true (total) differentiability requires a linear map approximating f with error o(‖h‖). Classic counterexample: f(x,y) = xy/(x²+y²).
