Partial differentiation | Total derivative | Multivariable calculus | Part 1

📘 Problem: Find the total derivative of the following function and also verify the result by direct substitution z= xy²+x²y where x=at and y=2at In this video, we solve this Partial Differentiation problem step by step, covering the essential concepts of Multivariable Calculus. Perfect for VTU 1BMATS101, 1BMATE101, 1BMATC101, 1BMATM101 (2025 Scheme) and BMATS101, BMATE101, BMATC101, BMATM101 (2022 Scheme) students. ⏩ Topics Covered: ✔️ Partial Differentiation basics ✔️ Application in Multivariable Calculus ✔️ Exam-focused solution 🔔 Support & More 👉 Join Now:    / @officialmathematicstutor   1️⃣ Find the total derivative of   z = xy² + x²y, x = at, y = 2at 🔗    • Partial differentiation | Total derivative...   2️⃣ Find du/dt for   u = x² + y² − z², x = eᵗ, y = eᵗcosh t, z = eᵗsinh t 🔗    • Partial differentiation | Total derivative...   3️⃣ If u = f(x/y, y/z, z/x), prove:   x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) = 0 🔗    • Partial differentiation | Total derivative...   4️⃣ If u = f(r, s, t), r = x/y, s = y/z, t = z/x, show:   xuₓ + yuᵧ + zu_z = 0 🔗    • Partial differentiation | Total derivative...   5️⃣ If u = f(x − y, y − z, z − x), prove:   uₓ + uᵧ + u_z = 0 🔗    • Partial differentiation | Total derivative...   6️⃣ If u = f(y − z, x − y, z − x), prove:   uₓ + uᵧ + u_z = 0 🔗    • Partial differentiation | Total derivative...   7️⃣ If u = f(2x − 3y, 3y − 4z, 4z − 2x), show:   (1/2)uₓ + (1/3)uᵧ + (1/4)u_z = 0 🔗    • Partial differentiation | Total derivative...   8️⃣ If u = f(2x − 3y, 3y − 4z, 4z − 2x), prove:   6uₓ + 4uᵧ + 3u_z = 0 🔗    • Partial differentiation | Total derivative...   9️⃣ If z = f(x, y), x = eᵘ sin v, y = eᵘ cos v, prove:   (zᵤ)² + (zᵥ)² = e²ᵘ[(zₓ)² + (zᵧ)²] 🔗    • Partial differentiation | Total derivative...   🔟 If z = f(x, y), x = r cosθ, y = r sinθ, prove:   (zₓ)² + (zᵧ)² = (zᵣ)² + (1/r²)(z_θ)² 🔗    • Partial differentiation | Total derivative...   #VTUMaths #TotalDerivative #PartialDifferentiation #MultivariableCalculus #EngineeringMathematics #ChainRule #HyperbolicFunctions #Calculus #Differentiation #MathsSolutions #VTU2025 #VTU2022 #1BMATS101 #1BMATE101 #1BMATC101 #1BMATM101 #BMATS101 #BMATE101 #BMATC101 #BMATM101 #MathematicsTutor