Question

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

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

Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle… 1st Edition

Chapter 3, Problem 64

Instant Answer

Solved by Expert Amy Jiang

05/27/2020

Step 1

We can use the quotient rule for differentiation, which states that the derivative of $\frac{u}{v}$ is $\frac{vu' - uv'}{v^2}$, where $u$ and $v$ are functions of $x$, and $u'$ and $v'$ are their respective derivatives. Here, $u=x$ and $v=x+2$. The derivative of Show more…

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

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Key Concepts

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.

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.

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