The functions s(x) and t(x) are differentiable. The function u(x) is defined as: u(x)= (s(x))/(t(x)) If s(6)= 7,s^(')(6)= 5,t(6)= 9, and t^(')(6)= 2, what is u^(')(6)? Simplify any fractions. u^(')(6)= _____

Quotient Rule for Derivatives: To find u'(6), we need to use the quotient rule for derivatives, which states that if u(x) = s(x)/t(x), then u'(x) = (s'(x)t(x) - s(x)t'(x))/(t(x))^2. We will apply this rule using the given values.Calculating the Numerator: First, we calculate the numerator of the quotient rule: s'(6)t(6) - s(6)t'(6) = 5*9 - 7*2.Numerator Calculation: Performing the calculation for the numerator: 5*9 - 7*2 = 45 - 14 = 31.Calculating the Denominator: Next, we calculate the denominator of the quotient rule: (t(6))^2 = 9^2.Denominator Calculation: Performing the calculation for the denominator: 9^2 = 81.Finding u'(6): Now, we can find u'(6) by dividing the numerator by the denominator: u'(6) = 31/81.Conclusion: We have found the derivative of u at x = 6, so we can conclude the solution.

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The functions $s(x)$ and $t(x)$ are differentiable. $\newline$ The function $u(x)$ is defined as: $u(x)= \frac{s(x)}{t(x)}$ $\newline$ If $s(6)= 7$ , $s'(6)= 5$ , $t(6)= 9$ , and $t'(6)= 2$ , what is $u'(6)$ ? $\newline$ Simplify any fractions. $\newline$ $u'(6)=$ _____

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Algebra 1

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Full solution

Q. The functions

s(x)

and

t(x)

are differentiable.

\newline

The function

u(x)

is defined as:

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

\newline

s(6)= 7

s'(6)= 5

t(6)= 9

, and

t'(6)= 2

, what is

u'(6)

\newline

Simplify any fractions.

\newline

u'(6)=

_____

Quotient Rule for Derivatives: To find $u'(6)$ , we need to use the quotient rule for derivatives, which states that if $u(x) = \frac{s(x)}{t(x)}$ , then $u'(x) = \frac{s'(x)t(x) - s(x)t'(x)}{(t(x))^2}$ . We will apply this rule using the given values.
Calculating the Numerator: First, we calculate the numerator of the quotient rule: $s'(6)t(6) - s(6)t'(6) = 5 \times 9 - 7 \times 2$ .
Numerator Calculation: Performing the calculation for the numerator: $5 \times 9 - 7 \times 2 = 45 - 14 = 31$ .
Calculating the Denominator: Next, we calculate the denominator of the quotient rule: $(t(6))^2 = 9^2$ .
Denominator Calculation: Performing the calculation for the denominator: $9^2 = 81$ .
Finding $u'(6)$ : Now, we can find $u'(6)$ by dividing the numerator by the denominator: $u'(6) = \frac{31}{81}$ .
Conclusion: We have found the derivative of $u$ at $x = 6$ , so we can conclude the solution.