The functions a(x) and b(x) are differentiable. The function w(x) is defined as: w(x)= (a(x))/(b(x)) If a(5)= 2,a^(')(5)= 1,b(5)= 4, and b^(')(5)= 2, what is w^(')(5)? Simplify any fractions. w^(')(5)=

Given function: We are given the function w(x) = a(x)/b(x) and we need to find the derivative of w at x = 5, denoted as w'(5). To do this, we will use the quotient rule for derivatives, which states that if w(x) = u(x)/v(x), then w'(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = a(x) and v(x) = b(x).Quotient rule for derivatives: We are given a(5) = 2, a'(5) = 1, b(5) = 4, and b'(5) = 2. We will substitute these values into the quotient rule formula.Substituting given values: Applying the quotient rule, we get w'(5) = (a'(5)b(5) - a(5)b'(5))/(b(5))^2.Calculating w'(5): Substituting the given values, we have w'(5) = (1*4 - 2*2)/(4)^2.Simplifying the result: Performing the calculations, we get w'(5) = (4 - 4)/(16).Simplifying the result, we find w'(5) = 0/16, which simplifies to 0.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The functions $a(x)$ and $b(x)$ are differentiable. The function $w(x)$ is defined as: $w(x)= \frac{a(x)}{b(x)}$ If $a(5)= 2$ , $a'(5)= 1$ , $b(5)= 4$ , and $b'(5)= 2$ , what is $w'(5)$ ? Simplify any fractions. $w'(5)=$

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

Q. The functions

a(x)

and

b(x)

are differentiable. The function

w(x)

is defined as:

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

a(5)= 2

a'(5)= 1

b(5)= 4

, and

b'(5)= 2

, what is

w'(5)

? Simplify any fractions.

w'(5)=

Given function: We are given the function $w(x) = \frac{a(x)}{b(x)}$ and we need to find the derivative of $w$ at $x = 5$ , denoted as $w'(5)$ . To do this, we will use the quotient rule for derivatives, which states that if $w(x) = \frac{u(x)}{v(x)}$ , then $w'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = a(x)$ and $v(x) = b(x)$ .
Quotient rule for derivatives: We are given $a(5) = 2$ , $a'(5) = 1$ , $b(5) = 4$ , and $b'(5) = 2$ . We will substitute these values into the quotient rule formula.
Substituting given values: Applying the quotient rule, we get $w'(5) = \frac{a'(5)b(5) - a(5)b'(5)}{(b(5))^2}$ .
Calculating $w'(5)$ : Substituting the given values, we have $w'(5) = (1 \cdot 4 - 2 \cdot 2)/(4)^2$ .
Simplifying the result: Performing the calculations, we get $w'(5) = \frac{(4 - 4)}{16}$ .
Simplifying the result: Performing the calculations, we get $w'(5) = \frac{4 - 4}{16}$ . Simplifying the result, we find $w'(5) = \frac{0}{16}$ , which simplifies to $0$ .