Partial Differentiation of Composite Function |u = f(x/y, y/z, z/x), u = f(x−y, y−z, z−x)

📘 Calculus & Multivariable Calculus | Partial Differentiation Problem Solutions 📝 *Problem 1:* If u = f(x/y, y/z, z/x), prove that x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) = 0. 🕘 *Problem 2 (at 9:00):* If u = f(x − y, y − z, z − x), prove that (∂u/∂x) + (∂u/∂y) + (∂u/∂z) = 0. 🎯 *Concepts Used:* ✔️ Partial Differentiation ✔️ Chain Rule for Multivariable Functions ✔️ Euler’s Theorem Type Relations ✔️ Simplification using Symmetry of Variables ✨ These problems beautifully illustrate how functions involving variable ratios or differences exhibit differential identities that simplify elegantly using multivariable calculus concepts. ✅ Very useful for Engineering Mathematics, Partial Differentiation, Multivariable Calculus, and VTU exam preparation. 💎 *Support Us:* Join this channel to support us & access exclusive perks 👇 🔗 / @officialmathematicstutor #Calculus #PartialDifferentiation #MultivariableCalculus #VTUMaths #EngineeringMathematics #1BMATS101 #1BMATC101 #1BMATM101 #1BMATE101

Partial differentiation | Total derivative | Multivariable calculus | Part 2

VTUM1|📌Partial differentiation very important problems u=log(x^3-y^3-z^3 −3xyz)

If u = f((y - x)/xy, (z - x)/xz) find x^2.∂u/∂x + y^2.∂u/∂y + z^2.∂u/∂z/MA3151/UNIT 3

If u = f(x-y, y-z, z-x) then show that ∂u/∂x + ∂u/∂y + ∂u/∂z = 0 / MA3151 /MATRICES AND CALCULUS

Assignment | If z = f(x, y) where x=u²-v², y = 2 uv ST (∂z/∂u)²+(∂z/∂v)²=4(u²+v²)[(∂z/∂x)²+(∂z/∂y)²]

Partial Differentiation of Composite Function | Multivariable Calculus | u = f(2x−3y, 3y−4z, 4z−2x)

Engg_Maths | If z=f(x, y) and x=r cos θ and y=r sin θ ST (∂z/∂x)²+(∂z/∂y)²=(∂z/∂r)²+(1/r²)(∂z/∂θ)²

Engg_Maths | Total Derivative | If u=f(x-y, y-x, z-x) Show That (∂u/∂x)+(∂u/∂y)+(∂u/∂z)=0

Partial Differentiation | Multivariable Calculus Find the Extreme Values of f(x, y)=x²+y²+6x−12

Engg_Maths | Total Derivative | If u = sin-1 (x - y), x= 3t, y = 4t³ Show That (du/dt) = 3÷√(1-t²)

#11 Jacobian in Hindi | if u= x+y/ x-y ,v=xy/(x-y)^2 examine u and v are functionally dependent

Jacobian Prove that u=xy+yz+zx,v=x^2+y^2+z^2, w=x+y+zare functionally dependent and find relation.

Ex of Jacobian If u= x^2-y^2 ,v=2xy then find (∂(x,y))/(∂(u,v))

Example on variable treated as a constant If u=mx+ny,v=nx-my

Composite If z=f(x,y)where x=e^u-e^(-v),y=e^(-u)-e^v then prove that (∂z/∂u-∂z/∂v)= x ∂z/∂x- y ∂z/∂y

Example 4 on PDE If u=tan⁡(y+ax)+(y-ax)^(3/2),prove that(∂^2 u)/(∂x^2)=a^2 (∂^2 u)/(∂y^2)[ Hindi ]

Partial Differentiation | Multivariable Calculus | z(x + y)=x²+y², (∂z/∂x−∂z/∂y)²=4(1−∂z/∂x −∂z/∂y)

Partial Differentiation | Multivariable Calculus | u = log(x³ + y³ + z³), ∂u/∂x + ∂u/∂y + ∂u/∂z = 0

Partial Differentiation | Multivariable Calculus | u=e^(ax+by) · f(ax–by), b(∂u/∂x)+a(∂u/∂y) = 2abu

If z = f(x, y) and x = eu + e-v , y = e-u – ev. Prove that 𝒅𝒛/𝒅𝒖 − 𝒅𝒛/𝒅𝒗 = x 𝒅𝒛/𝒅𝒙 – y 𝒅𝒛/𝒅𝒚 .