Question

Compute the difference​ quotient, StartFraction f (x plus h )minus f (x )Over h EndFraction f(x+h)−f(x) h​, for the function f (x )equals 3 x squaredf(x)=3x2​, and simplify.

Answer 1

The difference quotient for
`f(x)=3x^2` is
`3 h + 6 x`.

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

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

So, for the function
`f(x)=3x^2` the difference quotient is:

To find
`f(x+h)`, plug
`x+h` instead of
`x`

`f\left(x+h\right)=3 \left(h + x\right)^(2)`

Finally,

`(f\left(x+h\right)-f\left(x\right))/(h)=(\left(3 \left(h + x\right)^(2)\right)-\left(3 x^(2)\right))/(h)`

`(3\left(\left(h+x\right)^2-x^2\right))/(h) \\\\(3(h^2+2hx+x^2-x^2))/(h) \\\\(3h\left(h+2x\right))/(h)\\\\3h+6x`

The difference quotient for
`f(x)=3x^2` is
`3 h + 6 x`.