The functions s(x) and t(x) are differentiable. The function z(x) is defined as: z(x)= (s(x))/(t(x)) If s(4)= 7, s'(4)= 2, t(4)= 9, and t'(4)= -3, what is z'(4)? Simplify any fractions. z'(4)= _____

Apply Quotient Rule: To find z'(4), we need to use the quotient rule for differentiation, which states that if z(x) = s(x)/t(x), then z'(x) = (s'(x)t(x) - s(x)t'(x))/(t(x))^2. We will apply this rule using the given values.Calculate Numerator: First, we calculate the numerator of the quotient rule using the given derivatives and function values at x = 4: s'(4)t(4) - s(4)t'(4) = (2)(9) - (7)(-3).Perform Multiplication: Performing the multiplication, we get: 18 - (-21) = 18 + 21 = 39.Calculate Denominator: Next, we calculate the denominator of the quotient rule, which is (t(4))^2. Since t(4) = 9, we have (9)^2 = 81.Find z'(4): Now we can put together the numerator and the denominator to find z'(4): z'(4) = 39/81.Simplify Fraction: We can simplify the fraction 39/81 by dividing both the numerator and the denominator by 3: 39/81 = 13/27.

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The functions $s(x)$ and $t(x)$ are differentiable. $\newline$ The function $z(x)$ is defined as: $z(x)= \frac{s(x)}{t(x)}$ $\newline$ If $s(4)= 7$ , $s'(4)= 2$ , $t(4)= 9$ , and $t'(4)= -3$ , what is $z'(4)$ ? $\newline$ Simplify any fractions. $\newline$ $z'(4)=$ _____

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Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. The functions

s(x)

and

t(x)

are differentiable.

\newline

The function

z(x)

is defined as:

z(x)= \frac{s(x)}{t(x)}

\newline

s(4)= 7

s'(4)= 2

t(4)= 9

, and

t'(4)= -3

, what is

z'(4)

\newline

Simplify any fractions.

\newline

z'(4)=

_____

Apply Quotient Rule: To find $z'(4)$ , we need to use the quotient rule for differentiation, which states that if $z(x) = \frac{s(x)}{t(x)}$ , then $z'(x) = \frac{s'(x)t(x) - s(x)t'(x)}{(t(x))^2}$ . We will apply this rule using the given values.
Calculate Numerator: First, we calculate the numerator of the quotient rule using the given derivatives and function values at $x = 4$ : $s'(4)t(4) - s(4)t'(4) = (2)(9) - (7)(-3)$ .
Perform Multiplication: Performing the multiplication, we get: $18 - (-21) = 18 + 21 = 39$ .
Calculate Denominator: Next, we calculate the denominator of the quotient rule, which is $(t(4))^2$ . Since $t(4) = 9$ , we have $(9)^2 = 81$ .
Find $z'(4)$ : Now we can put together the numerator and the denominator to find $z'(4)$ : $z'(4) = \frac{39}{81}$ .
Simplify Fraction: We can simplify the fraction $\frac{39}{81}$ by dividing both the numerator and the denominator by $3$ : $\frac{39}{81} = \frac{13}{27}$ .