Partial differentiation | Total derivative | Multivariable calculus | Part 2

📘 Calculus & Multivariable Calculus | Total Derivative & Chain Rule Problem Solution 📝 Problem Statement: Find the *total derivative du/dt* for the function **u = x² + y² − z²**, where x = e^t, y = e^t cosh t, z = e^t sinh t, and *verify* the result by direct substitution (substitute x, y, z in u and differentiate directly with respect to t). 🎯 Concepts Used: ✔️ Total Derivative of a multivariable function ✔️ Partial Differentiation (∂u/∂x, ∂u/∂y, ∂u/∂z) ✔️ Chain Rule: du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt) + (∂u/∂z)(dz/dt) ✔️ Verification using direct substitution method ✔️ Hyperbolic function derivatives (cosh t, sinh t) ✨ This problem demonstrates two complementary methods to compute the derivative of a multivariable function with respect to a single variable and highlights the use of hyperbolic function identities. Essential practice for *VTU exams* and engineering mathematics. ✅ Very useful for Engineering Mathematics, Partial Differentiation, Multivariable Calculus, and VTU exam preparation. 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...   💎 *Support Us:* Join this channel to support us & access exclusive perks 👇 🔗    / @officialmathematicstutor   #Calculus #PartialDifferentiation #MultivariableCalculus #ChainRule #TotalDerivative #HyperbolicFunctions #VTU #EngineeringMathematics #BMATS201 #BMATE201 #BMATC201 #BMATM201 #1BMATS101 #1BMATE101 #1BMATC101 #1BMATM101 #Module1 #Module2