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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)=4 a(2)= 4 , a′(2)=5 a'(2)= 5 , b(2)=1 b(2)= 1 , and b′(2)=−3 b'(2)= -3 , 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)=4 a(2)= 4 , a′(2)=5 a'(2)= 5 , b(2)=1 b(2)= 1 , and b′(2)=−3 b'(2)= -3 , what is c′(2) c'(2) ? \newlineSimplify any fractions. \newlinec′(2)= c'(2)= _____
  1. Identify rule: Identify the rule for differentiating a quotient.\newlineThe quotient rule for differentiation states that if you have a function c(x)=a(x)b(x)c(x) = \frac{a(x)}{b(x)}, then 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}.
  2. Apply rule with values: Apply the quotient rule using the given values.\newlineWe have a(2)=4a(2) = 4, a′(2)=5a'(2) = 5, b(2)=1b(2) = 1, and b′(2)=−3b'(2) = -3. Plugging these into the quotient rule, we get:\newlinec′(2)=a′(2)b(2)−a(2)b′(2)(b(2))2c'(2) = \frac{a'(2)b(2) - a(2)b'(2)}{(b(2))^2}\newlinec′(2)=5⋅1−4⋅(−3)(1)2c'(2) = \frac{5\cdot 1 - 4\cdot (-3)}{(1)^2}
  3. Perform calculations: Perform the calculations.\newlinec′(2)=5−(−12)1c'(2) = \frac{5 - (-12)}{1}\newlinec′(2)=5+12c'(2) = 5 + 12\newlinec′(2)=17c'(2) = 17

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