Find and simplify the difference quotient (f(x+h) - f(x))/h for f(x) = 5x^2 - 3x + 9.

Step-by-step solution

1. 1

   Substitute x + h into the function. Replace every $x$, including the one inside the squared term:

   $f(x+h) = 5(x+h)^2 - 3(x+h) + 9 = 5x^2 + 10xh + 5h^2 - 3x - 3h + 9$

   Expanding $(x+h)^2$ as $x^2 + 2xh + h^2$ — not $x^2 + h^2$ — is the step that produces the $10xh$ term.

2. 2

   Subtract f(x). Every term without an $h$ must cancel, which is a built-in check on the expansion:

   $f(x+h) - f(x) = \left(5x^2 + 10xh + 5h^2 - 3x - 3h + 9\right) - \left(5x^2 - 3x + 9\right) = 10xh + 5h^2 - 3h$

   The constant $9$ and both $x$-only terms disappear, as they always do.

3. 3

   Factor h out of the numerator.

   $10xh + 5h^2 - 3h = h\left(10x + 5h - 3\right)$

4. 4

   Cancel the h. This is legitimate because the problem states $h \ne 0$:

   $\frac{h\left(10x + 5h - 3\right)}{h} = 10x + 5h - 3$

5. 5

   Check numerically and connect to the derivative. At $x = 1.3$, $h = 0.07$: the direct quotient $\tfrac{f(1.37) - f(1.3)}{0.07} = 10.35$, and the formula gives $10(1.3) + 5(0.07) - 3 = 10.35$. Letting $h \to 0$ leaves $10x - 3$, which is exactly $f'(x)$.