For the function $f (x, y) = 16 x y^{2}$ , find $\frac{f (x + h, y) - f (x, y)}{h}$ .
$\frac{f (x + h, y) - f (x, y)}{h} = ◻$

Answer

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Answer

Simplify the expression by canceling out the common factor of $h$ : $\frac{f (x + h, y) - f (x, y)}{h} = \frac{16 h y^{2}}{h} = 16 y^{2}$

Steps

Step 1 ：Substitute $f (x + h, y)$ into the numerator: $f (x + h, y) = 16 (x + h) y^{2}$

Step 2 ：Substitute $f (x, y)$ into the numerator: $f (x, y) = 16 x y^{2}$

Step 3 ：Substitute the values into the expression: $\frac{f (x + h, y) - f (x, y)}{h} = \frac{16 (x + h) y^{2} - 16 x y^{2}}{h}$

Step 4 ：Simplify the numerator: $16 (x + h) y^{2} - 16 x y^{2} = 16 x y^{2} + 16 h y^{2} - 16 x y^{2} = 16 h y^{2}$

Step 5 ：Substitute the simplified numerator back into the expression: $\frac{f (x + h, y) - f (x, y)}{h} = \frac{16 h y^{2}}{h}$

Step 6 ：Simplify the expression by canceling out the common factor of $h$ : $\frac{f (x + h, y) - f (x, y)}{h} = \frac{16 h y^{2}}{h} = 16 y^{2}$