Answer:
The difference quotient for [tex]f(x)=3x^2[/tex] is [tex]3 h + 6 x[/tex].
Step-by-step explanation:
The difference quotient is a formula that computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition the derivative and it is given by
[tex]\frac{f(x+h)-f(x)}{h}[/tex]
So, for the function [tex]f(x)=3x^2[/tex] the difference quotient is:
To find [tex]f(x+h)[/tex], plug [tex]x+h[/tex] instead of [tex]x[/tex]
[tex]f\left(x+h\right)=3 \left(h + x\right)^{2}[/tex]
Finally,
[tex]\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{\left(3 \left(h + x\right)^{2}\right)-\left(3 x^{2}\right)}{h}[/tex]
[tex]\frac{3\left(\left(h+x\right)^2-x^2\right)}{h} \\\\\frac{3(h^2+2hx+x^2-x^2)}{h} \\\\\frac{3h\left(h+2x\right)}{h}\\\\3h+6x[/tex]
The difference quotient for [tex]f(x)=3x^2[/tex] is [tex]3 h + 6 x[/tex].
