The functions a(x) and b(x) are differentiable. The function c(x) is defined as: c(x)= (a(x))/(b(x)) If a(2)= 5, a'(2)= 3, b(2)= 8, and b'(2)= -1, what is c'(2)? Simplify any fractions. c'(2)= _____

Given functions and derivatives: We are given the functions a(x) and b(x), and their derivatives at x = 2. We need to find the derivative of the function c(x) = a(x)/b(x) at x = 2. To do this, we will use the quotient rule for derivatives, which states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Applying the quotient rule: First, we will apply the quotient rule using the given values. We have a(2) = 5, a'(2) = 3, b(2) = 8, and b'(2) = -1. Plugging these into the quotient rule formula, we get c'(2) = (a'(2)b(2) - a(2)b'(2))/(b(2))^2.Substituting given values: Now, we substitute the given values into the formula: c'(2) = (3 * 8 - 5 * -1)/(8)^2.Performing the multiplication: Performing the multiplication, we get c'(2) = (24 + 5)/(64).Adding the numbers in the numerator: Adding the numbers in the numerator, we get c'(2) = 29/64.

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The functions $a(x)$ and $b(x)$ are differentiable. $\newline$ The function $c(x)$ is defined as: $c(x)= \frac{a(x)}{b(x)}$ $\newline$ If $a(2)= 5$ , $a'(2)= 3$ , $b(2)= 8$ , and $b'(2)= -1$ , what is $c'(2)$ ? $\newline$ Simplify any fractions. $\newline$ $c'(2)=$ _____

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Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. The functions

a(x)

and

b(x)

are differentiable.

\newline

The function

c(x)

is defined as:

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 5

a'(2)= 3

b(2)= 8

, and

b'(2)= -1

, what is

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

_____

Given functions and derivatives: We are given the functions $a(x)$ and $b(x)$ , and their derivatives at $x = 2$ . We need to find the derivative of the function $c(x) = \frac{a(x)}{b(x)}$ at $x = 2$ . To do this, we will use the quotient rule for derivatives, which states that if $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ .
Applying the quotient rule: First, we will apply the quotient rule using the given values. We have $a(2) = 5$ , $a'(2) = 3$ , $b(2) = 8$ , and $b'(2) = -1$ . Plugging these into the quotient rule formula, we get $c'(2) = \frac{a'(2)b(2) - a(2)b'(2)}{(b(2))^2}$ .
Substituting given values: Now, we substitute the given values into the formula: $c'(2) = \frac{(3 \times 8 - 5 \times -1)}{8^2}$ .
Performing the multiplication: Performing the multiplication, we get $c'(2) = \frac{24 + 5}{64}$ .
Adding the numbers in the numerator: Adding the numbers in the numerator, we get $c'(2) = \frac{29}{64}$ .