The functions c(x) and d(x) are differentiable. The function r(x) is defined as: r(x)= (c(x))/(d(x)) If c(6)= 7,c^(')(6)= 5,d(6)= 3, and d^(')(6)= -2, what is r^(')(6)? Simplify any fractions. r^(')(6)=

Given function: We are given the function r(x) = c(x)/d(x) and we need to find the derivative of r at x = 6, denoted as r'(6). To do this, we will use the quotient rule for derivatives, which states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Applying quotient rule: We have c(6) = 7, c'(6) = 5, d(6) = 3, and d'(6) = -2. Let's apply the quotient rule using these values: r'(6) = (c'(6)d(6) - c(6)d'(6))/(d(6))^2.Substituting values: Substitute the given values into the quotient rule formula: r'(6) = (5 * 3 - 7 * -2)/(3)^2.Performing multiplication: Perform the multiplication: r'(6) = (15 + 14)/(9).Adding numerator: Add the numbers in the numerator: r'(6) = 29/9.Final answer: The fraction 29/9 is already simplified, so this is our final answer.

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The functions $c(x)$ and $d(x)$ are differentiable. The function $r(x)$ is defined as: $r(x)= \frac{c(x)}{d(x)}$ If $c(6)= 7$ , $c'(6)= 5$ , $d(6)= 3$ , and $d'(6)= -2$ , what is $r'(6)$ ? Simplify any fractions. $r'(6)=$

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Algebra 1

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Q. The functions

c(x)

and

d(x)

are differentiable. The function

r(x)

is defined as:

r(x)= \frac{c(x)}{d(x)}

c(6)= 7

c'(6)= 5

d(6)= 3

, and

d'(6)= -2

, what is

r'(6)

? Simplify any fractions.

r'(6)=

Given function: We are given the function $r(x) = \frac{c(x)}{d(x)}$ and we need to find the derivative of $r$ at $x = 6$ , denoted as $r'(6)$ . To do this, we will use the quotient rule for derivatives, which states that if $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ .
Applying quotient rule: We have $c(6) = 7$ , $c'(6) = 5$ , $d(6) = 3$ , and $d'(6) = -2$ . Let's apply the quotient rule using these values: $\newline$ $r'(6) = \frac{c'(6)d(6) - c(6)d'(6)}{(d(6))^2}.$
Substituting values: Substitute the given values into the quotient rule formula: $r'(6) = \frac{(5 \cdot 3 - 7 \cdot -2)}{(3)^2}$ .
Performing multiplication: Perform the multiplication: $r'(6) = \frac{15 + 14}{9}$ .
Adding numerator: Add the numbers in the numerator: $r'(6) = \frac{29}{9}$ .
Final answer: The fraction $\frac{29}{9}$ is already simplified, so this is our final answer.