f(x+k,y)-f(x,y)/k when
f(x,y)=4x^2y
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.