Find the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$, where $$ h
eq 0 $$, for the function below. $$ f(x)=5 x^{2}-6 x+2 $$ Simplify your answer as much as possible. $$ \frac{f(x+h)-f(x)}{h}=\square $$

Question

Find the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$, where $$ h 
eq 0 $$, for the function below. $$ f(x)=5 x^{2}-6 x+2 $$ Simplify your answer as much as possible. $$ \frac{f(x+h)-f(x)}{h}=\square $$

UpStudy AI · Accepted Answer

We start by finding $$ f(x+h) $$ by replacing every $$ x $$ with $$ x+h $$ in the function:

$$
f(x) = 5x^2 - 6x + 2 \quad \Longrightarrow \quad f(x+h) = 5(x+h)^2 - 6(x+h) + 2.
$$

First, expand $$ (x+h)^2 $$:

$$
(x+h)^2 = x^2 + 2xh + h^2.
$$

Substitute back into $$ f(x+h) $$:

$$
\begin{aligned}
f(x+h) &= 5(x^2 + 2xh + h^2) - 6x - 6h + 2 \
       &= 5x^2 + 10xh + 5h^2 - 6x - 6h + 2.
\end{aligned}
$$

Now, form the difference quotient:

$$
\frac{f(x+h)-f(x)}{h} = \frac{\left[5x^2 + 10xh + 5h^2 - 6x - 6h + 2\right] - \left[5x^2 - 6x + 2\right]}{h}.
$$

Simplify the numerator by canceling like terms:

$$
\begin{aligned}
f(x+h) - f(x) &= \left(5x^2 + 10xh + 5h^2 - 6x - 6h + 2\right) - \left(5x^2 - 6x + 2\right) \
              &= 10xh + 5h^2 - 6h.
\end{aligned}
$$

Factor $$ h $$ from the numerator:

$$
10xh + 5h^2 - 6h = h(10x + 5h - 6).
$$

Now divide by $$ h $$ (noting that $$ h \neq 0 $$):

$$
\frac{h(10x + 5h - 6)}{h} = 10x + 5h - 6.
$$

Therefore, the simplified difference quotient is:

$$
\frac{f(x+h)-f(x)}{h} = 10x + 5h - 6.
$$
The difference quotient simplifies to $$ 10x + 5h - 6 $$.

Find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), where \( h \neq 0 \), for the function below. \( f(x)=5 x^{2}-6 x+2 \) Simplify your answer as much as possible. \( \frac{f(x+h)-f(x)}{h}=\square \)

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