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The functions a(x) a(x) and b(x) b(x) are differentiable. \newlineThe function c(x) c(x) is defined as: c(x)=a(x)b(x) c(x)= \frac{a(x)}{b(x)} \newlineIf a(2)=3 a(2)= 3 , a′(2)=4 a'(2)= 4 , b(2)=6 b(2)= 6 , and b′(2)=1 b'(2)= 1 , what is c′(2) c'(2) ? \newlineSimplify any fractions. \newlinec′(2)= c'(2)= _____

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Q. The functions a(x) a(x) and b(x) b(x) are differentiable. \newlineThe function c(x) c(x) is defined as: c(x)=a(x)b(x) c(x)= \frac{a(x)}{b(x)} \newlineIf a(2)=3 a(2)= 3 , a′(2)=4 a'(2)= 4 , b(2)=6 b(2)= 6 , and b′(2)=1 b'(2)= 1 , what is c′(2) c'(2) ? \newlineSimplify any fractions. \newlinec′(2)= c'(2)= _____
  1. Given Function: We are given the function c(x)=a(x)b(x)c(x) = \frac{a(x)}{b(x)} and we need to find the derivative of cc at x=2x = 2, denoted as c′(2)c'(2). To do this, we will use the quotient rule for derivatives, which states that if h(x)=f(x)g(x)h(x) = \frac{f(x)}{g(x)}, then h′(x)=f′(x)g(x)−f(x)g′(x)(g(x))2h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}. Here, f(x)=a(x)f(x) = a(x) and g(x)=b(x)g(x) = b(x).
  2. Apply Quotient Rule: Using the quotient rule, we can write the derivative of c(x)c(x) as c′(x)=a′(x)b(x)−a(x)b′(x)(b(x))2c'(x) = \frac{a'(x)b(x) - a(x)b'(x)}{(b(x))^2}. We need to evaluate this expression at x=2x = 2.
  3. Substitute and Evaluate: Substitute the given values into the derivative expression: c′(2)=a′(2)b(2)−a(2)b′(2)(b(2))2=(4⋅6−3⋅1)(6)2.c'(2) = \frac{a'(2)b(2) - a(2)b'(2)}{(b(2))^2} = \frac{(4 \cdot 6 - 3 \cdot 1)}{(6)^2}.
  4. Perform Calculations: Perform the calculations: c′(2)=24−336=2136.c'(2) = \frac{24 - 3}{36} = \frac{21}{36}.
  5. Simplify Fraction: Simplify the fraction: c′(2)=2136c'(2) = \frac{21}{36} can be simplified by dividing both the numerator and the denominator by 33, which gives us c′(2)=712c'(2) = \frac{7}{12}.

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