Answer:
[tex]=2x+h-2[/tex]
Step-by-step explanation:
When want to simplify:
[tex]\frac{f(x+h)-f(x)}{h}\text{ given } f(x)=x^2-2x[/tex]
So, let's substitute (x+h) into our f(x). This yields:
[tex]f(x+h)=(x+h)^2-2(x+h)[/tex]
Substitute this and f(x) into our above expression to obtain:
[tex]=\frac{(x+h)^2-2(x+h)-(x^2-2x)}{h}[/tex]
Simplify. First, square the first term:
[tex]=\frac{(x^2+2xh+h^2)-2(x+h)-(x^2-2x)}{h}[/tex]
Distribute the -2 for the second term:
[tex]=\frac{(x^2+2xh+h^2)-2x-2h-(x^2-2x)}{h}[/tex]
And distribute the negative for the last term:
[tex]=\frac{(x^2+2xh+h^2)-2x-2h-x^2+2x}{h}[/tex]
Combine like terms:
[tex]=\frac{(x^2-x^2)+(-2x+2x)+(2xh+h^2-2h)}{h}[/tex]
The first two expressions will cancel. This leaves us with:
[tex]=\frac{2xh+h^2-2h}{h}[/tex]
Factor out an h:
[tex]=\frac{h(2x+h-2)}{h}[/tex]
We can cancel the h. So, our final solution is:
[tex]=2x+h-2[/tex]
And we're done!
