• ∂f/∂x = 3x^2 y - 5 y^2 + 2 e^2x sin(y)
• ∂f/∂y = x^3 - 10 x y + e^2x cos(y)
• ∂²f/∂x∂y = 3x^2 - 10 y + 2 e^2x cos(y)
• ∂²f/∂y∂x = same (Clairaut holds).

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• Finding limits, derivatives, and integrals of vector functions.
• Computing velocity, speed, acceleration, and curvature.
• Parameterizing curves (lines, circles, helices).
• Finding unit tangent, normal, and binormal vectors (T, N, B).

f(x,y) = x y^3 + sin(x y). Compute f_xy and f_yx; verify equality.
Answer: f_x = y^3 + y cos(xy); f_xy = 3 y^2 + cos(xy) - x y sin(xy).
f_y = 3 x y^2 + x cos(xy); f_yx = 3 y^2 + cos(xy) - x y sin(xy). Equal.