Calculate and simplify the difference quotient for the function
y=x^3-125
y=x^3-125
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[ f(x+h) – f(x)] / h =
[ (x+h)³ – 125 – (x³ – 125)] / h =
[ x³ + 3x²h + 3xh² + h³ – 125 – x³ + 125] / h =
[ 3x²h + 3xh² + h³ ] / h =
3x² + 3xh + h²
and in the limit, as h → 0, that's just 3x²
[ (x+h)³ – 125 – (x³ – 125)] / h =
[ x³ + 3x²h + 3xh² + h³ – 125 – x³ + 125] / h =
[ 3x²h + 3xh² + h³ ] / h =
3x² + 3xh + h²
and in the limit, as h → 0, that's just 3x²