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)= 5,p^(')(3)= 2,q(3)= 7, and q^(')(3)= -3, what is r^(')(3)? Simplify any fractions. r^(')(3)= _____

Identify Rule: Identify the rule for differentiating a quotient. The quotient rule states that if r(x) = p(x)/q(x), then r'(x) = (p'(x)q(x) - p(x)q'(x))/(q(x))^2.Apply Quotient Rule: Apply the quotient rule to find r'(3). Using the values given: p(3) = 5, p'(3) = 2, q(3) = 7, and q'(3) = -3, we can substitute these into the quotient rule formula. r'(3) = (p'(3)q(3) - p(3)q'(3))/(q(3))^2 r'(3) = (2*7 - 5*(-3))/(7)^2Perform Calculations: Perform the calculations. r'(3) = (14 + 15)/(49) r'(3) = 29/49

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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 $p(3)= 5$ , $p'(3)= 2$ , $q(3)= 7$ , and $q'(3)= -3$ , 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)= 5

p'(3)= 2

q(3)= 7

, and

q'(3)= -3

, what is

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

_____

Identify Rule: Identify the rule for differentiating a quotient. $\newline$ The quotient rule states that if $r(x) = \frac{p(x)}{q(x)}$ , then $r'(x) = \frac{p'(x)q(x) - p(x)q'(x)}{(q(x))^2}$ .
Apply Quotient Rule: Apply the quotient rule to find $r'(3)$ . Using the values given: $p(3) = 5$ , $p'(3) = 2$ , $q(3) = 7$ , and $q'(3) = -3$ , we can substitute these into the quotient rule formula. $r'(3) = \frac{p'(3)q(3) - p(3)q'(3)}{(q(3))^2}$ $r'(3) = \frac{2\cdot7 - 5\cdot(-3)}{7^2}$
Perform Calculations: Perform the calculations. $\newline$ $r'(3) = \frac{14 + 15}{49}$ $\newline$ $r'(3) = \frac{29}{49}$