In multivariable calculus, a partial derivative measures how a function changes when only one of its variables varies, while the others are held fixed. Mixed partials—derivatives taken with respect to different variables—are especially useful for understanding curvature and optimization.

Step 1: Differentiate with respect to x

Take the derivative of f(x, y) = 3x²y – 2xy with respect to x, treating y as a constant:

∂f/∂x = 6xy – 2y

Step 2: Differentiate the result with respect to y

Now differentiate ∂f/∂x = 6xy – 2y with respect to y, treating x as constant:

∂²f/(∂y∂x) = 6x – 2

Step 3: Verify symmetry of mixed partials

Compute ∂²f/(∂x∂y) by differentiating ∂f/∂y = 3x² – 2x with respect to x:

∂²f/(∂x∂y) = 6x – 2

Since ∂²f/(∂y∂x) = ∂²f/(∂x∂y), the mixed partials are equal, confirming Clairaut’s theorem for this smooth function.

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