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)= 4,a^(')(2)= 5,b(2)= 1, and b^(')(2)= -3, what is c^(')(2)? Simplify any fractions. c^(')(2)= _____

Identify rule: Identify the rule for differentiating a quotient. The quotient rule for differentiation states that if you have a function c(x) = a(x)/b(x), then c'(x) = (a'(x)b(x) - a(x)b'(x))/(b(x))^2.Apply rule with values: Apply the quotient rule using the given values. We have a(2) = 4, a'(2) = 5, b(2) = 1, and b'(2) = -3. Plugging these into the quotient rule, we get: c'(2) = (a'(2)b(2) - a(2)b'(2))/(b(2))^2 c'(2) = (5*1 - 4*(-3))/(1)^2Perform calculations: Perform the calculations. c'(2) = (5 - (-12))/1 c'(2) = 5 + 12 c'(2) = 17

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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 $a(2)= 4$ , $a'(2)= 5$ , $b(2)= 1$ , and $b'(2)= -3$ , 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)= 4

a'(2)= 5

b(2)= 1

, and

b'(2)= -3

, what is

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

_____

Identify rule: Identify the rule for differentiating a quotient. $\newline$ The quotient rule for differentiation states that if you have a function $c(x) = \frac{a(x)}{b(x)}$ , then $c'(x) = \frac{a'(x)b(x) - a(x)b'(x)}{(b(x))^2}$ .
Apply rule with values: Apply the quotient rule using the given values. $\newline$ We have $a(2) = 4$ , $a'(2) = 5$ , $b(2) = 1$ , and $b'(2) = -3$ . Plugging these into the quotient rule, we get: $\newline$ $c'(2) = \frac{a'(2)b(2) - a(2)b'(2)}{(b(2))^2}$ $\newline$ $c'(2) = \frac{5\cdot 1 - 4\cdot (-3)}{(1)^2}$
Perform calculations: Perform the calculations. $\newline$ $c'(2) = \frac{5 - (-12)}{1}$ $\newline$ $c'(2) = 5 + 12$ $\newline$ $c'(2) = 17$