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The functions a(x) a(x) and b(x) b(x) are differentiable. The function w(x) w(x) is defined as: w(x)=a(x)b(x) w(x)= \frac{a(x)}{b(x)} If a(5)=2 a(5)= 2 , a′(5)=1 a'(5)= 1 , b(5)=4 b(5)= 4 , and b′(5)=2 b'(5)= 2 , what is w′(5) w'(5) ? Simplify any fractions. w′(5)= w'(5)=

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Q. The functions a(x) a(x) and b(x) b(x) are differentiable. The function w(x) w(x) is defined as: w(x)=a(x)b(x) w(x)= \frac{a(x)}{b(x)} If a(5)=2 a(5)= 2 , a′(5)=1 a'(5)= 1 , b(5)=4 b(5)= 4 , and b′(5)=2 b'(5)= 2 , what is w′(5) w'(5) ? Simplify any fractions. w′(5)= w'(5)=
  1. Given function: We are given the function w(x)=a(x)b(x)w(x) = \frac{a(x)}{b(x)} and we need to find the derivative of ww at x=5x = 5, denoted as w′(5)w'(5). To do this, we will use the quotient rule for derivatives, which states that if w(x)=u(x)v(x)w(x) = \frac{u(x)}{v(x)}, then w′(x)=u′(x)v(x)−u(x)v′(x)(v(x))2w'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}. Here, u(x)=a(x)u(x) = a(x) and v(x)=b(x)v(x) = b(x).
  2. Quotient rule for derivatives: We are given a(5)=2a(5) = 2, a′(5)=1a'(5) = 1, b(5)=4b(5) = 4, and b′(5)=2b'(5) = 2. We will substitute these values into the quotient rule formula.
  3. Substituting given values: Applying the quotient rule, we get w′(5)=a′(5)b(5)−a(5)b′(5)(b(5))2w'(5) = \frac{a'(5)b(5) - a(5)b'(5)}{(b(5))^2}.
  4. Calculating w′(5)w'(5): Substituting the given values, we have w′(5)=(1⋅4−2⋅2)/(4)2w'(5) = (1 \cdot 4 - 2 \cdot 2)/(4)^2.
  5. Simplifying the result: Performing the calculations, we get w′(5)=(4−4)16w'(5) = \frac{(4 - 4)}{16}.
  6. Simplifying the result: Performing the calculations, we get w′(5)=4−416w'(5) = \frac{4 - 4}{16}. Simplifying the result, we find w′(5)=016w'(5) = \frac{0}{16}, which simplifies to 00.

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