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Partial Differentiation and Applications

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Examples for Rules of Partial Differentiation:

Derivative of Sum:

$δ/{δx} (u±v) = {δu}/{δx}±{δv}/{δx} $

Derivative of Product:
$δ/{δx} (uv) = u {δv}/{δx} + v {δu}/{δx}$

Derivative of Quotient:


$δ/{δx} (u/v) = {v {δu}/{δx} – u {δv}/{δx}}/{v^2}$

If k is constant, then
$δ/{δx} (ku) = k {δu}/{δx} $

Derivative of constant is 0

$δ/{δx} [f(x,y,z)]^n = n[f(x,y,z)]^{n-1} {δf}/{δx}$

$δ/{δx} [1/{f(x,y,z)}] = {-1}/{[f(x,y,z)]^2} {δf}/{δx}$

$δ/{δx} [√{f(x,y,z)}] = 1/{2√{f(x,y,z)} {δf}/{δx}}$

$ δ/{δx} [log f(x,y,z)] = 1/{f(x,y,z)} {δf}/{δx} $

$ δ/{δx} [a^{f(x,y,z)}] = a^{f(x,y,z)} log a {δf}/{δx} $

$ δ/{δx} [sin f(x,y,z)] = cos f(x,y,z) {δf}/{δx} $

$ δ/{δx} [cos f(x,y,z)] = -sin f(x,y,z) {δf}/{δx} $

$ δ/{δx} [tan f(x,y,z)] = sec^2 f(x,y,z) {δf}/{δx} $

$ δ/{δx} [cot f(x,y,z)] = -cosec^2 f(x,y,z) {δf}/{δx} $

$ δ/{δx} [cosec f(x,y,z)] = -cosec f(x,y,z) cot f(x,y,z) {δf}/{δx} $

$ δ/{δx} [sec f(x,y,z)] = sec f(x,y,z) tan f(x,y,z) {δf}/{δx} $

$ δ/{δx} [sin^{-1} f(x,y,z)] = 1/{√{1 – [f(x,y,z)]^2}} {δf}/{δx} $

$ δ/{δx} [tan^{-1} f(x,y,z)] = 1/{1 + [f(x,y,z)]^2} {δf}/{δx} $

Derivative of Composite function (or function of a function )

$ {δz}/{δx} = {dz}/{dr} {δr}/{δx} or {df}/{dr} {δr}/{δx} or f'(r) {δr}/{δx} $

Examples of Partial Derivatives of Higher Order:

$ δ/{δx} ({δz}/{δx}) = {δ^2 z}/{δx^2}$ or ${δ^2 f}/{δx^2}$ or $Z_{xx}$ or $f_{xx}$

$ δ/{δy} ({δz}/{δx}) = {δ^2 z}/{δy δx}$ or ${δ^2 f}/{δy δx}$ or $Z_{yx}$ or $f_{yx}$

$ {δ^3 z}/{δx^3} = δ/{δx} [δ/{δx}({δz}/{δx})] $

$ {δ^3 z}/{δx^2 δy} = δ/{δx} [δ/{δx}({δz}/{δy})] $

${δ^3 z}/{δx δy δx} = δ/{δx}[δ/{δy} ({δz}/{δx})]$

${δ^2 z}/{δx δy} = {δ^2 z}/{δy δx}$

u -> r -> x,y
${δu}/{δx} = {du}/{dr} {δr}/{δx}$ , ${δu}/{δy} = {du}/{dr} {δr}/{δy}$

$r^2 = x^2 + y^2$
$ {δr}/{δx} = x/r$ ${δr}/{δy} = y/r$

Variable to be treated as constant:
x=r cosθ, y=r sinθ. Then $r^2 = x^2 + y^2$
$({δr}/{δx})_θ$ means partial derivative of r with respect to x, treating θ as constant in a relation expressing r as a function of x and θ only.

Homogenous functions of two variables
$z = x^n f(y/x)$ , $z = y^n f(x/y)$

Euler’s Theorem:

$x {δz}/{δx} + y {δz}/{δy} = nz$

$x^2 {δ^2 z}/{δx^2} + 2xy {δ^2 z}/{δx δy} + y^2 {δ^2 z}/{δy^2} = n(n-1) z$

$ x {δu}/{δx} + y {δu}/{δy} = n {f(u)}/{f'(u)} $

$x^2 {δ^2 z}/{δx^2} + 2xy {δ^2 z}/{δx δy} + y^2 {δ^2 z}/{δy^2} = g(u)[g'(u) – 1] $

u -> x,y,z -> t
Total derivative: ${du}/{dt} = {δu}/{δx} {dx}/{dt} + {δu}/{δy} {dy}/{dt} + {δu}/{δz} {dz}/{dt}$

Total differential: $du = {δu}/{δx} dz + {δu}/{δy} dy + {δu}/{δz} dz$

Differentiation of implicit functions

${dy}/{dx} = – {{δf}/{δx}}/{{δf}/{δy}} = – p/q$

$q^3 {d^2 y}/{dx^2} = |\table r,s,p;s,t,q;p,q,o|$