Question

72 - 73. first and second derivatives find $f(x)$ and $f(x)$. 72. $f(x)=\frac{x}{x + 2}$ 73. $f(x)=\frac{x^{2}-7x}{x + 1}$

Explanation:

Step1: Use quotient - rule for first derivative

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. For $f(x)=\frac{x}{x + 2}$, where $u = x$, $u'=1$, $v=x + 2$, $v'=1$.

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Step2: Use quotient - rule for second derivative

Now, for $y = f'(x)=\frac{2}{(x + 2)^{2}}=\frac{2}{(x + 2)^{2}}$, where $u = 2$, $u'=0$, $v=(x + 2)^{2}$, $v'=2(x + 2)$.

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Answer:

For $f(x)=\frac{x}{x + 2}$, $f'(x)=\frac{2}{(x + 2)^{2}}$, $f''(x)=-\frac{4}{(x + 2)^{3}}$

Now for $f(x)=\frac{x^{2}-7x}{x + 1}$:

Step1: Use quotient - rule for first derivative

Let $u=x^{2}-7x$, $u'=2x-7$, $v=x + 1$, $v'=1$.

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Step2: Use quotient - rule for second derivative

Let $u=x^{2}+2x - 7$, $u'=2x+2$, $v=(x + 1)^{2}$, $v'=2(x + 1)$.

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