The functions m(x) and n(x) are differentiable. The function o(x) is defined as: o(x)= (m(x))/(n(x)) If m(7)= 2,m^(')(7)= -1,n(7)= 4, and n^(')(7)= 8, what is o^(')(7)? Simplify any fractions. o^(')(7)= _____

Identify Rule: Identify the rule for differentiating a quotient. The quotient rule for differentiation states that if you have a function o(x) = m(x)/n(x), then its derivative o'(x) is given by: o'(x) = (m'(x) * n(x) - m(x) * n'(x)) / (n(x))^2.Apply Rule: Apply the quotient rule using the given values. We are given m(7) = 2, m'(7) = -1, n(7) = 4, and n'(7) = 8. Plugging these values into the quotient rule, we get: o'(7) = ((-1) * 4 - 2 * 8) / (4)^2.Perform Calculations: Perform the calculations. Now we calculate the numerator and the denominator separately: Numerator: (-1) * 4 - 2 * 8 = -4 - 16 = -20. Denominator: (4)^2 = 16. So, o'(7) = -20 / 16.Simplify Fraction: Simplify the fraction. The fraction -20 / 16 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4: o'(7) = (-20 / 4) / (16 / 4) = -5 / 4.

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The functions $m(x)$ and $n(x)$ are differentiable. $\newline$ The function $o(x)$ is defined as: $o(x)= \frac{m(x)}{n(x)}$ $\newline$ If $m(7)= 2$ , $m'(7)= -1$ , $n(7)= 4$ , and $n'(7)= 8$ , what is $o'(7)$ ? $\newline$ Simplify any fractions. $\newline$ $o'(7)=$ _____

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Algebra 1

Compare linear and exponential growth

Full solution

Q. The functions

m(x)

and

n(x)

are differentiable.

\newline

The function

o(x)

is defined as:

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

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

, and

n'(7)= 8

, what is

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

_____

Identify Rule: Identify the rule for differentiating a quotient. $\newline$ The quotient rule for differentiation states that if you have a function $o(x) = \frac{m(x)}{n(x)}$ , then its derivative $o'(x)$ is given by: $\newline$ $o'(x) = \frac{m'(x) \cdot n(x) - m(x) \cdot n'(x)}{(n(x))^2}$ .
Apply Rule: Apply the quotient rule using the given values. $\newline$ We are given $m(7) = 2$ , $m'(7) = -1$ , $n(7) = 4$ , and $n'(7) = 8$ . Plugging these values into the quotient rule, we get: $\newline$ $o'(7) = ((-1) \cdot 4 - 2 \cdot 8) / (4)^2$ .
Perform Calculations: Perform the calculations. $\newline$ Now we calculate the numerator and the denominator separately: $\newline$ Numerator: $(-1) \times 4 - 2 \times 8 = -4 - 16 = -20$ . $\newline$ Denominator: $(4)^2 = 16$ . $\newline$ So, $o'(7) = \frac{-20}{16}$ .
Simplify Fraction: Simplify the fraction. $\newline$ The fraction $-\frac{20}{16}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ : $\newline$ o'( $7$ ) = $\left(-\frac{20}{4}\right) / \left(\frac{16}{4}\right) = -\frac{5}{4}$ .