Question

simplify f(x+h) - f(x) / h

f(x)=x^2+x

Answer 1

ah the difference quotient so beloved by math teachers. only hard part is figuring out what
\[f(x+h)\] means. replace x by x+h to get
\[f(x+h)=(x+h)^2+(x+h)=x^2+2xh+h^2+x+h\]

Answer 2

then subtract and divide. everything with out an h in it will disappear from the numerator
\[\frac{f(x+h)-f(x)}{h}=\frac{x^2+2xh+h^2+x+h-(x^2+x)}{h}\]
\[=frac{x^2+2xh+h^2+x+h-x^2-x}{h}=\frac{2xh+h+h^2}{h}\]

Answer 3

now factor and cancel to get
\[\frac{f(x+h)-f(x)}{h}=\frac{h(2x+1+h)}{h}=2x+1+h\]

Answer 4

ah great thx

Answer 5

typo above should be
\[=\frac{x^2+2xh+h^2+x+h-x^2-x}{h}=\frac{2xh+h+h^2}{h}\]

Answer 6

yw