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Find and simplify the difference quotient
(f(x+h)-f(x))/(h),h!=0 for thogiven function.
5)
f(x)=x^(2)+7x+3

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}, h \neq 0$ for thogiven function. $\newline$ $5$ ) $f(x)=x^{2}+7 x+3$

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Relative maxima or minima

Full solution

Q. Find and simplify the difference quotient

\frac{f(x+h)-f(x)}{h}, h \neq 0

for thogiven function.

\newline

5

)

f(x)=x^{2}+7 x+3

Substitute $f(x)$ and $f(x+h)$ : Step $1$ : Substitute $f(x)$ and $f(x+h)$ into the difference quotient formula. $\newline$ $f(x) = x^2 + 7x + 3$ $\newline$ $f(x+h) = (x+h)^2 + 7(x+h) + 3$
Expand $f(x+h)$ : Step $2$ : Expand $f(x+h)$ .
$f(x+h) = (x^2 + 2xh + h^2) + (7x + 7h) + 3$
= $x^2 + 2xh + h^2 + 7x + 7h + 3$
Calculate $f(x+h) - f(x)$ : Step $3$ : Calculate $f(x+h) - f(x)$ .
$f(x+h) - f(x) = (x^2 + 2xh + h^2 + 7x + 7h + 3) - (x^2 + 7x + 3)$
$= 2xh + h^2 + 7h$
Divide to simplify: Step $4$ : Divide by $h$ to simplify the difference quotient. $\newline$ $(f(x+h) - f(x)) / h = (2xh + h^2 + 7h) / h$ $\newline$ $= 2x + h + 7$

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