For the function {eq}f(x)=\frac{x^2}{3+x}{/eq}, find {eq}f"(x){/eq} then {eq}f"(0){/eq} then {eq}f"(2){/eq}.

Question:

For the function {eq}f(x)=\frac{x^2}{3+x}{/eq}, find {eq}f"(x){/eq} then {eq}f"(0){/eq} then {eq}f"(2){/eq}.

Quotient Rule in Differentiation

The quotient rule in differentiation states that {eq}\displaystyle \frac{d}{dx} \frac{f(x)}{g(x)} =\frac{g(x)f'(x)-f(x)g'(x)}{{(g(x))}^2} {/eq}

We need to know that {eq}\displaystyle \frac{d}{dx} x^n = nx^{n-1}, \frac{d}{dx} k=0 {/eq} where {eq}\displaystyle k {/eq} is a constant.

{eq}\displaystyle f''(x) {/eq} is obtained by differentiating the first derivative of the function {eq}\displaystyle f(x) {/eq} again.{eq}\displaystyle \frac{d}{dx}f'(x)=f''(x) {/eq}

Answer and Explanation: 1

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Given {eq}\displaystyle f(x)= \frac{x^2}{x+3} {/eq}

Differentiating, we get

{eq}\displaystyle \frac{d}{dx}f(x) =\frac{d}{dx}...

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When to Use the Quotient Rule for Differentiation

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Chapter 8 / Lesson 8
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The quotient rule can be used for differentiation when taking the derivative of a function divided by another function. Gain a better understanding of when to use the quotient rule and explore some examples in this lesson.


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