Difference Quotient Calculator

What is a Difference Quotient?

The Difference Quotient formula is $${f(x+h)-f(x)\over h}$$.

The Difference Quotient can be looked at in 2 different ways.
In Algebra we learn that the slope m of a line passing through the two points (x1,y1) and (x2,y2) is $m= {y2-y1 \over x2-x1}$
Moving on to Calculus, we generalize this concept to Functions. We compute the so called Secant Line Slope as the slope through the two points (x,f(x)) and (x+h,f(x+h)) that lie on the Function f(x).

Thus, the formula for the Difference Quotient is ${f(x+h)-f(x) \over (x+h)-x }$ which simplifies to the above Difference Quotient Formula ${f(x+h)-f(x)\over h}$ .

The slope of the red line in the above image is the secant line slope which is exactly what the Difference Quotient finds.

How do I find the Difference Quotients? How do I simplify the Difference Quotient?

Example1 (simple): If f(x)=2x+5 then its Difference Quotient is ${f(x+h)-f(x) \over h } ={2(x+h)+5 - (2x+5) \over h} = {2h\over h} = 2$ which is the slope between any 2 points on the line 2x+5.
That was not complicated ;-)

Example2 (more advanced): If f(x)=x^2+5x then its Difference Quotient is ${f(x+h)-f(x) \over h} = {(x+h)^2+5(x+h) - (x^2+(5x)) \over h} =$ ${(x^2+2xh+h^2)+5x+5h - (x^2+5x) \over h} = {2xh+h^2+5h \over h} = 2x+h$
That was a bit more complicated. Now try the above Difference Quotient Calculator a few times.

The Difference Quotient in Calculus: In Calculus, we extend the idea of a Difference Quotient. It is used to find the slope at one point instead of between two points. To accomplish this, we take the limit as h->0 of (f(x+h)-f(x))/h which moves the "dummy" point (x+h, f(x+h)) towards point (x,f(x)).
By finding the slope between those two approaching points we actually find the slope at (x,f(x)). This is called "Finding the Slope at a Point by using the Limit of a Difference Quotient of a Function" in Calculus, quite a mouthful.

Example: If f(x)=x^2 then ${f(x+h)-f(x) \over h} = {(x+h)^2 - x^2 \over h } = {2hx +h^2\over h} = 2x + h = 2x$ as limit h->0

This tells us that the slope at any point on the graph of f(x)=x^2 is 2x thanks to the Difference Quotient and taking the Limit h->0.

For instance, the slope of f(x)=x^2 at x=5 is computed as 2*5 = 10.

Here is another Difference Quotient calculator that you may like: https://calculator-online.net/difference-quotient-calculator/

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