Question

Find $f^{\prime}(x)$ and $f^{\prime \prime}(x).$ $$f(x)=\frac{x}{x+2}$$

   Find $f^{\prime}(x)$ and $f^{\prime \prime}(x).$
$$f(x)=\frac{x}{x+2}$$

Calculus: Early Transcendentals

Calculus: Early Transcendentals

William Briggs, Lyle… 3rd Edition

Chapter 3, Problem 72

Instant Answer

Solved by Expert Alex Roush

06/13/2020

Step 1

We use the quotient rule which states that the derivative of $\frac{u}{v}$ is $\frac{vu' - uv'}{v^2}$, where $u$ is the numerator, $v$ is the denominator, and the primes denote derivatives. Here, $u = x$, $v = x+2$, $u' = 1$, and $v' = 1$. So, we have: \[f'(x) = Show more…

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Find $f^{\prime}(x)$ and $f^{\prime \prime}(x).$ $$f(x)=\frac{x}{x+2}$$

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Key Concepts

Quotient Rule

The quotient rule is a method for differentiating functions that are expressed as the ratio of two differentiable functions. It states that if a function is given by f(x) = g(x)/h(x), then its derivative f'(x) is [g'(x)h(x) - g(x)h'(x)] divided by [h(x)]². This rule is essential for finding the first derivative of rational functions.

Second Derivative

The second derivative is the derivative of the first derivative and provides insight into the concavity and inflection points of a function. When the first derivative is a rational function, finding the second derivative typically involves applying differentiation rules, such as the quotient rule or product rule, again to differentiate the first derivative.

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Transcript

00:01 Okay, so our task is to find the first and second derivatives of this function.

00:04 So the first derivative is clearly an example of the quotient rule because we have the quotient of two functions.

00:10 So f prime of x is going to be the bottom function times the derivative of the top function, which is just one since this is linear, minus the top function times the derivative of the bottom function, which is also linear with the slope of one.

00:24 All divided by the bottom function squared.

00:27 Okay, this is x minus x and the x is can.

00:32 And we're just left with the 2.

00:34 So the first derivative is thus 2 divided by x plus 2 quantity squared.

00:41 Now finding the second derivative, i'm assuming we don't know the chain rule, which means we actually can't just differentiate this function without expanding it.

00:50 So i'm going to rewrite this.

00:53 Maybe i'll do it over here.

00:54 I'm going to rewrite this as x squared plus twice the product of these guys, 4x, plus 4.

01:02 So that we can now differentiate a second time.

01:10 Okay, so the second derivative is the derivative of the first derivative.

01:14 So we're differentiating this function now.

01:18 And again, we have a quotient of two functions...

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