The functions m(x) and n(x) are differentiable. The function z(x) is defined as: z(x)= (m(x))/(n(x)) If m(7)= 3,m^(')(7)= -2,n(7)= 5, and n^(')(7)= 3, what is z^(')(7)? Simplify any fractions. z^(')(7)=

Given function: We are given the function z(x) = m(x)/n(x) and we need to find the derivative of z at x = 7, which is z'(7). To do this, we will use the quotient rule for derivatives, which states that if z(x) = u(x)/v(x), then z'(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = m(x) and v(x) = n(x).Quotient rule for derivatives: We are given m(7) = 3, m'(7) = -2, n(7) = 5, and n'(7) = 3. We will substitute these values into the quotient rule formula.Substituting given values: Using the quotient rule, we get z'(7) = (m'(7)n(7) - m(7)n'(7))/(n(7))^2.Performing multiplication: Substituting the given values, we get z'(7) = ((-2)(5) - (3)(3))/(5)^2.Adding numerator: Performing the multiplication, we get z'(7) = (-10 - 9)/(25).Final result: Adding the numbers in the numerator, we get z'(7) = (-19)/(25).The fraction cannot be simplified further, so z'(7) = -19/25.

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

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Algebra 1

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Q. The functions

m(x)

and

n(x)

are differentiable. The function

z(x)

is defined as:

z(x)= \frac{m(x)}{n(x)}

m(7)= 3

m'(7)= -2

n(7)= 5

, and

n'(7)= 3

, what is

z'(7)

? Simplify any fractions.

z'(7)=

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