Maths WorkshopMaths problems with full solutions

Partial derivatives

25 problems · Partial derivatives, gradients and the chain rule.

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Ex.257Evaluating partial derivatives at a point

#partial derivatives #evaluation #point

Partial derivatives

Ex.258First partial derivatives

#partial derivatives #definition #computation

Partial derivatives

Ex.380Partial derivatives and the gradient

#partial derivatives #gradient #chain rule

Partial derivatives

Ex.522Directional derivative

#directional derivative #gradient #direction

Partial derivatives

Ex.523Euler's theorem for homogeneous functions

#homogeneous functions #Euler #partial derivatives

Partial derivatives

Ex.524The gradient

#gradient #vector #direction

Partial derivatives

Ex.771The chain rule for a composition

#chain rule #composition #directional derivative

Partial derivatives

Ex.1079General chain rule

#chain rule #composition #multivariable

Partial derivatives

Ex.1080Clairaut's theorem

#Clairaut #Schwarz #mixed partials

Partial derivatives

Ex.1082The differential

#differential #linearization #approximation

Partial derivatives

Ex.1083Second-order partial derivatives

#higher order #Schwarz #second derivatives

Partial derivatives

Ex.1084The Laplacian

#Laplacian #harmonic functions #second derivatives

Partial derivatives

Ex.1085Finding critical points

#critical points #stationary #gradient

Partial derivatives

Ex.1086Schwarz's theorem and a counterexample

#Schwarz theorem #mixed partials #symmetry

Partial derivatives

Ex.1087Tangent plane to a graph

#tangent plane #differentiability #graph

Partial derivatives

Ex.1498Is a function $C^1$ on $\mathbb{R}^2$?

#continuity #partial derivatives #class C1

Partial derivatives

Ex.1795Differentiability of a piecewise function

#differentiability #continuity #piecewise

Partial derivatives

Ex.1797Implicit differentiation

#implicit differentiation #Implicit Function Theorem

Partial derivatives

Ex.1798Inverse function theorem (illustration)

#inverse function theorem #Jacobian #local invertibility

Partial derivatives

Ex.1799Jacobian of a composition

#Jacobian #composition #chain rule

Partial derivatives

Ex.1800Level curves and gradients

#level curves #gradient #orthogonality

Partial derivatives

Ex.1801Mean value inequality

#mean value #inequality #gradient

Partial derivatives

Ex.1802Taylor expansion of order 2

#Taylor expansion #multivariable #approximation

Partial derivatives

Ex.2185Harmonic conjugate

#harmonic conjugate #Cauchy-Riemann #complex

Partial derivatives

Ex.2186Partial derivatives exist but function is not differentiable

#differentiability #counterexample #partial derivatives

Partial derivatives