The functions p(x) and q(x) are differentiable. The function r(x) is defined as: r(x)= (p(x))/(q(x)) If p(3)= 2,p^(')(3)= 4,q(3)= 6, and q^(')(3)= 9, what is r^(')(3)? Simplify any fractions. r^(')(3)= _____

Given function: We are given the function r(x) = p(x)/q(x) and we need to find the derivative of r at x = 3, denoted as r'(3). To do this, we will use the quotient rule for derivatives, which states that if r(x) = p(x)/q(x), then r'(x) = (p'(x)q(x) - p(x)q'(x))/(q(x))^2.Quotient rule for derivatives: We have the values p(3) = 2, p'(3) = 4, q(3) = 6, and q'(3) = 9. Let's plug these values into the quotient rule formula to find r'(3). r'(3) = (p'(3)q(3) - p(3)q'(3))/(q(3))^2 r'(3) = (4*6 - 2*9)/(6)^2Plugging in values: Now, let's perform the calculations: r'(3) = (24 - 18)/(36) r'(3) = 6/36 r'(3) = 1/6

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The functions $p(x)$ and $q(x)$ are differentiable. $\newline$ The function $r(x)$ is defined as: $r(x)= \frac{p(x)}{q(x)}$ $\newline$ If $p(3)= 2$ , $p'(3)= 4$ , $q(3)= 6$ , and $q'(3)= 9$ , what is $r'(3)$ ? $\newline$ Simplify any fractions. $\newline$ $r'(3)=$ _____

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Algebra 1

Compare linear and exponential growth

Full solution

Q. The functions

p(x)

and

q(x)

are differentiable.

\newline

The function

r(x)

is defined as:

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

, and

q'(3)= 9

, what is

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

_____

Given function: We are given the function $r(x) = \frac{p(x)}{q(x)}$ and we need to find the derivative of $r$ at $x = 3$ , denoted as $r'(3)$ . To do this, we will use the quotient rule for derivatives, which states that if $r(x) = \frac{p(x)}{q(x)}$ , then $r'(x) = \frac{p'(x)q(x) - p(x)q'(x)}{(q(x))^2}$ .
Quotient rule for derivatives: We have the values $p(3) = 2$ , $p'(3) = 4$ , $q(3) = 6$ , and $q'(3) = 9$ . Let's plug these values into the quotient rule formula to find $r'(3)$ . $\newline$ $r'(3) = \frac{p'(3)q(3) - p(3)q'(3)}{(q(3))^2}$ $\newline$ $r'(3) = \frac{4 \cdot 6 - 2 \cdot 9}{6^2}$
Plugging in values: Now, let's perform the calculations: $\newline$ r'( $3$ ) = $\frac{24 - 18}{36}$ $\newline$ r'( $3$ ) = $\frac{6}{36}$ $\newline$ r'( $3$ ) = $\frac{1}{6}$