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)= -1, q(3)= 4, and q'(3)= 2, what is r'(3)? Simplify any fractions. r'(3)= _____

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Apply Quotient Rule: To find r'(3), we need to use the quotient rule for differentiation, which states that if r(x) = p(x)/q(x), then r'(x) = (p'(x)q(x) - p(x)q'(x))/(q(x))^2. We will apply this rule using the given values for p(3), p'(3), q(3), and q'(3).Calculate Numerator: First, we calculate the numerator of the quotient rule: p'(3)q(3) - p(3)q'(3). Substituting the given values, we get: (-1)(4) - (2)(2).Calculate Denominator: Performing the multiplication, we find the numerator to be: -4 - 4 = -8.Find r'(3): Next, we calculate the denominator of the quotient rule: (q(3))^2. Substituting the given value for q(3), we get: (4)^2.Simplify Fraction: Calculating the square, we find the denominator to be: 16.Now, we can find r'(3) by dividing the numerator by the denominator: r'(3) = -8/16.Simplifying the fraction, we get: r'(3) = -1/2.

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 $p(3)= 2$ , $p'(3)= -1$ , $q(3)= 4$ , and $q'(3)= 2$ , what is $r'(3)$ ? $\newline$ Simplify any fractions. $\newline$ $r'(3)=$ _____

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