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)= 3,a^(')(2)= 4,b(2)= 6, and b^(')(2)= 1, what is c^(')(2)? Simplify any fractions. c^(')(2)= _____

Given Function: We are given the function c(x) = a(x)/b(x) and we need to find the derivative of c at x = 2, denoted as c'(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. Here, f(x) = a(x) and g(x) = b(x).Apply Quotient Rule: Using the quotient rule, we can write the derivative of c(x) as c'(x) = (a'(x)b(x) - a(x)b'(x))/(b(x))^2. We need to evaluate this expression at x = 2.Substitute and Evaluate: Substitute the given values into the derivative expression: c'(2) = (a'(2)b(2) - a(2)b'(2))/(b(2))^2 = (4*6 - 3*1)/(6)^2.Perform Calculations: Perform the calculations: c'(2) = (24 - 3)/(36) = 21/36.Simplify Fraction: Simplify the fraction: c'(2) = 21/36 can be simplified by dividing both the numerator and the denominator by 3, which gives us c'(2) = 7/12.

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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 $a(2)= 3$ , $a'(2)= 4$ , $b(2)= 6$ , and $b'(2)= 1$ , what is $c'(2)$ ? $\newline$ Simplify any fractions. $\newline$ $c'(2)=$ _____

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Algebra 1

Compare linear and exponential 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)= 3

a'(2)= 4

b(2)= 6

, and

b'(2)= 1

, what is

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

_____

Given Function: We are given the function $c(x) = \frac{a(x)}{b(x)}$ and we need to find the derivative of $c$ at $x = 2$ , denoted as $c'(2)$ . To do this, we will use the quotient rule for derivatives, which states that if $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ . Here, $f(x) = a(x)$ and $g(x) = b(x)$ .
Apply Quotient Rule: Using the quotient rule, we can write the derivative of $c(x)$ as $c'(x) = \frac{a'(x)b(x) - a(x)b'(x)}{(b(x))^2}$ . We need to evaluate this expression at $x = 2$ .
Substitute and Evaluate: Substitute the given values into the derivative expression: $c'(2) = \frac{a'(2)b(2) - a(2)b'(2)}{(b(2))^2} = \frac{(4 \cdot 6 - 3 \cdot 1)}{(6)^2}.$
Perform Calculations: Perform the calculations: $c'(2) = \frac{24 - 3}{36} = \frac{21}{36}.$
Simplify Fraction: Simplify the fraction: $c'(2) = \frac{21}{36}$ can be simplified by dividing both the numerator and the denominator by $3$ , which gives us $c'(2) = \frac{7}{12}$ .