# Difference Quotient Solver

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Difference quotient solver is used in the derivative. Dividing the function difference from the difference of the points is called as difference quotient.  The difference quotient is used to calculate the slope of a secant line.  It is also defined as a slope of secant line passing through two points (x, f(x)) and (x + h, f(x + h)). The slope of a secant line is calculated as m = (f(x + h) – f(x)) / (x + h) – x. by simplifying this we get slope = (f(x + h) – f(x)) / h.

Problem 1: Find the difference quotient of function f(x) = 7x + 4.

Solution: Given function is f(x) = 7x + 4

=> So f(x + h) = 7(x + h) + 4 = 7x +7h + 4

=> Now f(x + h) - f(x) = 7x + 7h + 4 - (7x + 4) = 7x + 7h + 4 - 7x - 4 = 7h

=> We have (f(x + h) – f(x)) / h = 7h / h = 7

=> The difference quotient of function f(x) = 7x + 4 is 7.

Problem 2: Find the difference quotient of the function f(x) =4x^2 +2x – 1.

Solution: Given function is f(x) =4x^2 +2x – 1

=> We can write f(x + h) = 4(x + h) ^2+ 2(x + h) – 1

=> (f (x + h) – f(x)) / h = (4(x + h) ^2+ 2(x + h) – 1 – (4x^2 +2x – 1))/ h

= (4(x^2 + h^2 + 2xh) + 2x + 2h – 1 – 4x^2 +2x – 1)/ h

= (4x^2 +4 h^2 + 8xh + 2x + 2h – 1 – 4x^2 -2x + 1)/ h = 4h + 8x + 2

=> So the difference quotient of this function is 4h + 8x + 2.