The functions m(x) and n(x) are differentiable. The function o(x) is defined as: o(x)= (m(x))/(n(x)) If m(0)= 1, m'(0)= 4, n(0)= 3, and n'(0)= -2, what is o'(0)? Simplify any fractions. o'(0)= _____

Apply Quotient Rule: To find o'(0), we need to use the quotient rule for differentiation, which states that if o(x) = m(x)/n(x), then o'(x) = (m'(x)n(x) - m(x)n'(x))/(n(x))^2. We will apply this rule at x = 0.Calculate Numerator: First, we calculate the numerator of the quotient rule at x = 0: m'(0)n(0) - m(0)n'(0) = (4)(3) - (1)(-2).Calculate Denominator: Performing the multiplication, we get: (4)(3) - (1)(-2) = 12 - (-2) = 12 + 2 = 14.Find o'(0): Next, we calculate the denominator of the quotient rule at x = 0: (n(0))^2 = (3)^2.Squaring n(0), we get: (3)^2 = 9.Now we can put together the numerator and denominator to find o'(0): o'(0) = 14/9.

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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 $m(0)= 1$ , $m'(0)= 4$ , $n(0)= 3$ , and $n'(0)= -2$ , what is $o'(0)$ ? $\newline$ Simplify any fractions. $\newline$ $o'(0)=$ _____

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Algebra 2

Even and odd functions

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(0)= 1

m'(0)= 4

n(0)= 3

, and

n'(0)= -2

, what is

o'(0)

\newline

Simplify any fractions.

\newline

o'(0)=

_____

Apply Quotient Rule: To find $o'(0)$ , we need to use the quotient rule for differentiation, which states that if $o(x) = \frac{m(x)}{n(x)}$ , then $o'(x) = \frac{m'(x)n(x) - m(x)n'(x)}{(n(x))^2}$ . We will apply this rule at $x = 0$ .
Calculate Numerator: First, we calculate the numerator of the quotient rule at $x = 0$ : $m'(0)n(0) - m(0)n'(0) = (4)(3) - (1)(-2)$ .
Calculate Denominator: Performing the multiplication, we get: $(4)(3) - (1)(-2) = 12 - (-2) = 12 + 2 = 14$ .
Find $o'(0)$ : Next, we calculate the denominator of the quotient rule at $x = 0$ : $(n(0))^2 = (3)^2$ .
Find $o'(0)$ : Next, we calculate the denominator of the quotient rule at $x = 0$ : $(n(0))^2 = (3)^2$ . Squaring $n(0)$ , we get: $(3)^2 = 9$ .
Find $o'(0)$ : Next, we calculate the denominator of the quotient rule at $x = 0$ : $(n(0))^2 = (3)^2$ . Squaring $n(0)$ , we get: $(3)^2 = 9$ . Now we can put together the numerator and denominator to find $o'(0)$ : $o'(0) = \frac{14}{9}$ .