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## To find the expression f(x+k, y) - f(x, y) / k, where f(x, y) = 4x^2y, we will substitute the given function into the expression.

Step 1: Find f(x+k, y) by substituting (x+k) for x in the function f(x, y).

f(x+k, y) = 4(x+k)^2y

Step 2: Expand the squared term.

f(x+k, y) = 4(x^2 + 2kx + k^2)y

= 4x^2y + 8kxy + 4k^2y

Step 3: Find f(x, y).

f(x, y) = 4x^2y

Step 4: Substitute the values of f(x+k, y) and f(x, y) into the expression f(x+k, y) - f(x, y) / k.

(f(x+k, y) - f(x, y)) / k = (4x^2y + 8kxy + 4k^2y - 4x^2y) / k

Step 5: Simplify the expression by canceling out like terms.

(f(x+k, y) - f(x, y)) / k = (8kxy + 4k^2y) / k

Step 6: Divide each term in the numerator by k.

(f(x+k, y) - f(x, y)) / k = 8xy + 4ky

Therefore, the expression f(x+k, y) - f(x, y) / k is equal to 8xy + 4ky.