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The functions s(x) s(x) and t(x) t(x) are differentiable. \newlineThe function u(x) u(x) is defined as: u(x)=s(x)t(x) u(x)= \frac{s(x)}{t(x)} \newlineIf s(10)=5 s(10)= 5 , s′(10)=3 s'(10)= 3 , t(10)=2 t(10)= 2 , and t′(10)=−4 t'(10)= -4 , what is u′(10) u'(10) ? \newlineSimplify any fractions. \newlineu′(10)= u'(10)= _____

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Q. The functions s(x) s(x) and t(x) t(x) are differentiable. \newlineThe function u(x) u(x) is defined as: u(x)=s(x)t(x) u(x)= \frac{s(x)}{t(x)} \newlineIf s(10)=5 s(10)= 5 , s′(10)=3 s'(10)= 3 , t(10)=2 t(10)= 2 , and t′(10)=−4 t'(10)= -4 , what is u′(10) u'(10) ? \newlineSimplify any fractions. \newlineu′(10)= u'(10)= _____
  1. Apply Quotient Rule: To find u′(x)u'(x), we need to use the quotient rule for derivatives, which states that if u(x)=s(x)t(x)u(x) = \frac{s(x)}{t(x)}, then u′(x)=s′(x)t(x)−s(x)t′(x)(t(x))2u'(x) = \frac{s'(x)t(x) - s(x)t'(x)}{(t(x))^2}. We will apply this rule to find u′(10)u'(10).
  2. Calculate Numerator: First, we calculate the numerator of the quotient rule: s′(10)t(10)−s(10)t′(10)s'(10)t(10) - s(10)t'(10). Plugging in the given values, we get 3×2−5×(−4)3 \times 2 - 5 \times (-4).
  3. Calculate Denominator: Performing the multiplication, we have 6−(−20)6 - (-20), which simplifies to 6+206 + 20.
  4. Find u′(10)u'(10): Adding the numbers together, we get 2626 for the numerator of the derivative at x=10x = 10.
  5. Divide Numerator by Denominator: Next, we calculate the denominator of the quotient rule: (t(10))2(t(10))^2. Plugging in the given value, we get (2)2(2)^2.
  6. Simplify Fraction: Squaring the number 22, we get 44 for the denominator of the derivative at x=10x = 10.
  7. Simplify Fraction: Squaring the number 22, we get 44 for the denominator of the derivative at x=10x = 10.Now, we divide the numerator by the denominator to find u′(10)u'(10): 264\frac{26}{4}.
  8. Simplify Fraction: Squaring the number 22, we get 44 for the denominator of the derivative at x=10x = 10.Now, we divide the numerator by the denominator to find u′(10):264u'(10): \frac{26}{4}.Simplifying the fraction 264\frac{26}{4}, we get 6.56.5 or 132\frac{13}{2} as the value of u′(10)u'(10).

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