Answer
(a)
$f^{(1)}(x)=\dfrac{d}{dx}(\cot(x^{2})) = -2 x \csc^2(x^2)$
$f^{(2)}(x)=(8x^2cot(x^2)−2)csc^2(x^2)$
$f^{(3)}(x)=(24xcot(x^2)−32x^3cot^2(x^2))csc^2(x^2)−16x^3csc^4(x^2)$
(b)
$f^{(1)}(x)=\dfrac{3x^2}{2\sqrt{x^3+1}}$
$f^{(2)}=\dfrac{3x^4+12x}{4(x^3+1)^{\frac{3}{2}}}$
$f^{(3)}(x)=\dfrac{-3x^6-60x^3+24}{8(x^3+1)^{\frac{5}{2}}}$
Work Step by Step
(a)
The given function is $f (x) = \cot(x^{2})$
Use computer algebra system to find $f^{(k)}(x)$ for $k=1,2,3$.
For $k=1$
$f^{(1)}(x)=\dfrac{d}{dx}(\cot(x^{2})) = -2 x \csc^2(x^2)$
For $k=2$
$f^{(2)}(x)=(8x^2cot(x^2)−2)csc^2(x^2)$
For $k=3$
$f^{(3)}(x)=(24xcot(x^2)−32x^3cot^2(x^2))csc^2(x^2)−16x^3csc^4(x^2)$
(b)
The given function is $f (x) = \sqrt{x^3+1}$
Use computer algebra system to find $f^{(k)}(x)$ for $k=1,2,3$.
For $k=1$
$f^{(1)}(x)=\dfrac{3x^2}{2\sqrt{x^3+1}}$
For $k=2$
$f^{(2)}=\dfrac{3x^4+12x}{4(x^3+1)^{\frac{3}{2}}}$
For $k=3$
$f^{(3)}(x)=\dfrac{-3x^6-60x^3+24}{8(x^3+1)^{\frac{5}{2}}}$
