Question

compute the first two derivatives of $f(x)=sqrt{5x - 1}$. a. $f(x)=$ b. $f(x)=$

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\sqrt{5x - 1}=(5x - 1)^{\frac{1}{2}}$.

Step2: Find the first - derivative using the chain rule

The chain rule is $(u^n)^\prime=nu^{n - 1}\cdot u^\prime$. Let $u = 5x-1$ and $n=\frac{1}{2}$. Then $u^\prime=5$. So $f^\prime(x)=\frac{1}{2}(5x - 1)^{-\frac{1}{2}}\cdot5=\frac{5}{2\sqrt{5x - 1}}$.

Step3: Find the second - derivative using the quotient rule

The quotient rule is $(\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 5$ and $v = 2\sqrt{5x - 1}=2(5x - 1)^{\frac{1}{2}}$. $u^\prime = 0$ and $v^\prime=2\times\frac{1}{2}(5x - 1)^{-\frac{1}{2}}\times5=\frac{5}{\sqrt{5x - 1}}$. Then $f^{\prime\prime}(x)=\frac{0\times2\sqrt{5x - 1}-5\times\frac{5}{\sqrt{5x - 1}}}{4(5x - 1)}=-\frac{25}{4(5x - 1)^{\frac{3}{2}}}$.

Answer:

a. $\frac{5}{2\sqrt{5x - 1}}$
b. $-\frac{25}{4(5x - 1)^{\frac{3}{2}}}$