Find the derivative of the function. 1) f(x)=arcsin⁡(x−1) 2) f(x)=arccos⁡x 3) f(x)=arctan⁡ex 4) f(x)=sin⁡(arccos⁡x)

Given: 1) f(x)=arcsin⁡(x−1)diffirentiate w. r. to xf′(x)=ddx[arcsin⁡(x−1)]=1x−(x2+x+2x)f′(x)=1−x2−2x2) f(x)=arccos⁡xdifferentiate w.r to xf′(x)=ddx(arccos⁡x)=−11−(x)2ddx(x)=−11−x⋅12xf′(x)=−12x1−xf(x)=arctan⁡exdifferentiate w.r. to xf′(x)=ddx(arctan⁡ex)=11+(ex)2ddxexf′(x)=ex1+e2x4) f(x)=arcsin⁡x+arccos⁡xdifferentiate w.r. to xf′(x)=ddx(arcsin⁡x)+ddx(arccos⁡x)=11−x2−11−x2f′(x)=0

Question: "Find the derivative of the function. 1) f(x)=arcsin⁡(x−1)..."

arenceabigns

Answered question

2021-06-08

Find the derivative of the function.
1) $f (x) = \arcsin (x - 1)$
2) $f (x) = \arccos \sqrt{x}$
3) $f (x) = \arctan e^{x}$
4) $f (x) = \sin (\arccos x)$

Answer & Explanation

pierretteA

Skilled2021-06-09Added 102 answers

Given: 1) $f (x) = \arcsin (x - 1)$
diffirentiate w. r. to x
$f^{'} (x) = \frac{d}{d x} [\arcsin (x - 1)]$
$= \frac{1}{\sqrt{x - (x^{2} + x + 2 x})}$
$f^{'} (x) = \frac{1}{\sqrt{- x^{2} - 2 x}}$
2) $f (x) = \arccos \sqrt{x}$
differentiate w.r to x
$f^{'} (x) = \frac{d}{d x} (\arccos \sqrt{x})$
$= - \frac{1}{\sqrt{1 - (\sqrt{x})^{2}}} \frac{d}{d x} (\sqrt{x})$
$= \frac{- 1}{\sqrt{1 - x}} \cdot \frac{1}{2 \sqrt{x}}$
$f^{'} (x) = \frac{- 1}{2 \sqrt{x} \sqrt{1 - x}}$
$f (x) = \arctan e^{x}$
differentiate w.r. to x
$f^{'} (x) = \frac{d}{d x} (\arctan e^{x})$
$= \frac{1}{1 + (e^{x})^{2}} \frac{d}{d x} e^{x}$
$f^{'} (x) = \frac{e^{x}}{1 + e^{2 x}}$
4) $f (x) = \arcsin x + \arccos x$
differentiate w.r. to x
$f^{'} (x) = \frac{d}{d x} (\arcsin x) + \frac{d}{d x} (\arccos x)$
$= \frac{1}{\sqrt{1 - x^{2}}} - \frac{1}{\sqrt{1 - x^{2}}}$
$f^{'} (x) = 0$

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