Explained: "Find the derivatives of the functions f(s)=s−1s+1"

College
Calculus and Analysis
Differential Calculus
Derivatives

Clifland

Answered question

2021-10-16

Find the derivatives of the functions

f (s) = \sqrt{s} - \frac{1}{\sqrt{s}} + 1

Answer & Explanation

grbavit

Skilled2021-10-17Added 109 answers

Given,
$f (s) = \sqrt{s} - \frac{1}{\sqrt{s}} + 1$
$\Rightarrow f (s) = s^{\frac{1}{2}} - \frac{1}{s^{\frac{1}{2}}} + 1$
$\Rightarrow f (s) = s^{\frac{1}{2}} - s^{- \frac{1}{2}} + 1$
Differentiating with respect to s, we get
$f^{'} (s) = \frac{1}{2} s^{\frac{1}{2} - 1} - (- \frac{1}{2} s^{- \frac{1}{2} - 1}) + 0$

$[∵ \frac{d (x^{n})}{d x} = n \cdot x^{n - 1}; \frac{d (k)}{d x} = 0 w h e r e k i s a n y c o n s t a n t]$
$\Rightarrow f^{'} (s) = \frac{1}{2} s^{- \frac{1}{2}} + \frac{1}{2} s^{- \frac{3}{2}}$
$\Rightarrow f^{'} (s) = \frac{1}{2 s^{\frac{1}{2}}} + \frac{1}{2 s^{\frac{3}{2}}}$
$\Rightarrow f^{'} (s) = \frac{1}{2 \sqrt{s}} + \frac{1}{2 \sqrt{3} {s}}$

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