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The functions m(x) m(x) and n(x) n(x) are differentiable. The function z(x) z(x) is defined as: z(x)=m(x)n(x) z(x)= \frac{m(x)}{n(x)} If m(7)=3 m(7)= 3 , m′(7)=−2 m'(7)= -2 , n(7)=5 n(7)= 5 , and n′(7)=3 n'(7)= 3 , what is z′(7) z'(7) ? Simplify any fractions. z′(7)= z'(7)=

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Q. The functions m(x) m(x) and n(x) n(x) are differentiable. The function z(x) z(x) is defined as: z(x)=m(x)n(x) z(x)= \frac{m(x)}{n(x)} If m(7)=3 m(7)= 3 , m′(7)=−2 m'(7)= -2 , n(7)=5 n(7)= 5 , and n′(7)=3 n'(7)= 3 , what is z′(7) z'(7) ? Simplify any fractions. z′(7)= z'(7)=
  1. Given function: We are given the function z(x)=m(x)n(x)z(x) = \frac{m(x)}{n(x)} and we need to find the derivative of zz at x=7x = 7, which is z′(7)z'(7). To do this, we will use the quotient rule for derivatives, which states that if z(x)=u(x)v(x)z(x) = \frac{u(x)}{v(x)}, then z′(x)=u′(x)v(x)−u(x)v′(x)(v(x))2z'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}. Here, u(x)=m(x)u(x) = m(x) and v(x)=n(x)v(x) = n(x).
  2. Quotient rule for derivatives: We are given m(7)=3m(7) = 3, m′(7)=−2m'(7) = -2, n(7)=5n(7) = 5, and n′(7)=3n'(7) = 3. We will substitute these values into the quotient rule formula.
  3. Substituting given values: Using the quotient rule, we get z′(7)=m′(7)n(7)−m(7)n′(7)(n(7))2z'(7) = \frac{m'(7)n(7) - m(7)n'(7)}{(n(7))^2}.
  4. Performing multiplication: Substituting the given values, we get z′(7)=((−2)(5)−(3)(3))(5)2z'(7) = \frac{((-2)(5) - (3)(3))}{(5)^2}.
  5. Adding numerator: Performing the multiplication, we get z′(7)=−10−925z'(7) = \frac{-10 - 9}{25}.
  6. Final result: Adding the numbers in the numerator, we get z′(7)=−1925z'(7) = \frac{-19}{25}.
  7. Final result: Adding the numbers in the numerator, we get z′(7)=−1925z'(7) = \frac{-19}{25}. The fraction cannot be simplified further, so z′(7)=−1925z'(7) = -\frac{19}{25}.

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