The difference quotient can be found by the formula:
[tex]\frac{f(x+h)-f(x)}{h}[/tex]
Where h is not 0, because it would be a division by 0.
In this case, we have the function:
[tex]f(x)=x^2-7x+1[/tex]
Then using the forumla:
[tex]\frac{f(x+h)-f(x)}{h}=\frac{(x+h)^2-7(x+h)+1-(x^2-7x+1)}{h}[/tex]
The next step is to solve the binomial squared, and do a distributive property.
We can solve the squared binomial using:
[tex](a+b)^2=a^2+2ab+b^2[/tex]
Then:
[tex]\frac{(x+h)^2-7(x+h)+1-(x^2-7x+1)}{h}=\frac{x^2+2xh+h^2-7x-7h+1-x^2+7x-1}{h}[/tex]
Here we can see that there are several terms equal but with different sign. We can cancel out the terms x², 7x and 1
[tex]\frac{x^2+2xh+h^2-7x-7h+1-x^2+7x-1}{h}=\frac{x^2-x^2-7x+7x+1-1+2hx+7h+h^2}{h}=\frac{h^2+2hx-7h}{h}[/tex]
Now we can factor out the h in the numerator:
[tex]\frac{h(h+2x-7)}{h}[/tex]
Which cancels out with the denominator, and we get:
[tex]\frac{f(x+h)-f(x)}{h}=2x-7+h[/tex]
And that's the result of the problem.
