The functions g(x) and h(x) are differentiable. The function i(x) is defined as: i(x)= (g(x))/(h(x)) If g(8)= 4,g^(')(8)= 3,h(8)= 2, and h^(')(8)= -4, what is i^(')(8)? Simplify any fractions. i^(')(8)= _____

Apply Quotient Rule: Use the quotient rule for differentiation: (u/v)^(') = (u^(')v - uv^('))/(v^2). Here, u = g(x) and v = h(x).Calculate Derivatives: Calculate u^(')(8) and v^(')(8) using the given derivatives: u^(')(8) = g^(')(8) = 3, v^(')(8) = h^(')(8) = -4.Calculate Function Values: Calculate u(8) and v(8) using the given function values: u(8) = g(8) = 4, v(8) = h(8) = 2.Plug into Formula: Plug the values into the quotient rule formula: i^(')(8) = (3 * 2 - 4 * -4) / (2^2).Simplify Numerator: Simplify the numerator: i^(')(8) = (6 + 16) / 4.Simplify Fraction: Simplify the fraction: i^(')(8) = 22 / 4.Reduce to Lowest Terms: Reduce the fraction to lowest terms: i^(')(8) = 11 / 2.

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The functions $g(x)$ and $h(x)$ are differentiable. $\newline$ The function $i(x)$ is defined as: $i(x)= \frac{g(x)}{h(x)}$ $\newline$ If $g(8)= 4$ , $g'(8)= 3$ , $h(8)= 2$ , and $h'(8)= -4$ , what is $i'(8)$ ? $\newline$ Simplify any fractions. $\newline$ $i'(8)=$ _____

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Algebra 1

Compare linear and exponential growth

Full solution

Q. The functions

g(x)

and

h(x)

are differentiable.

\newline

The function

i(x)

is defined as:

i(x)= \frac{g(x)}{h(x)}

\newline

g(8)= 4

g'(8)= 3

h(8)= 2

, and

h'(8)= -4

, what is

i'(8)

\newline

Simplify any fractions.

\newline

i'(8)=

_____

Apply Quotient Rule: Use the quotient rule for differentiation: $(\frac{u}{v})^{'} = \frac{u^{'}v - uv^{'}}{v^2}$ . Here, $u = g(x)$ and $v = h(x)$ .
Calculate Derivatives: Calculate $u'(8)$ and $v'(8)$ using the given derivatives: $u'(8) = g'(8) = 3$ , $v'(8) = h'(8) = -4$ .
Calculate Function Values: Calculate $u(8)$ and $v(8)$ using the given function values: $u(8) = g(8) = 4$ , $v(8) = h(8) = 2$ .
Plug into Formula: Plug the values into the quotient rule formula: $i^{'}(8) = \frac{(3 \times 2 - 4 \times -4)}{(2^2)}$ .
Simplify Numerator: Simplify the numerator: $i^{'}(8) = \frac{(6 + 16)}{4}$ .
Simplify Fraction: Simplify the fraction: $i^{(8)} = \frac{22}{4}$ .
Reduce to Lowest Terms: Reduce the fraction to lowest terms: $i^{(8)} = \frac{11}{2}$ .