Collection of Maths Problems

Derivatives of trig. and inv trig. functions

Task number: 3012

• Variant 1

  \( \sqrt{\sin x} \)

• Result

  \(\displaystyle f'=\frac{\cos x}{2 \sqrt{\sin x}} \), \(D_f=\langle0,\pi\rangle+2k\pi\), \(D_{f'}=(0,\pi)+2k\pi\)

• Variant 2

  \( \cos\sqrt{x-1} \)

• Result

  \(\displaystyle f'=\frac{-\sin\sqrt{x-1}}{2\sqrt{x-1}} \), \(D_f=\langle1,\infty)\), \(D_{f'}=(1,\infty)\)

• Variant 3

  \( \cos^3(2x) \)

• Result

  \(\displaystyle f'=-6\cos^2(2x)\sin(2x) \), \(D_f=D_{f'}=\mathbb R\)

• Variant 4

  \( \sin\sqrt{x+2}\)

• Result

  \(\displaystyle f'=\frac{\cos\sqrt{x+2}}{2\sqrt{x+2}} \), \(D_f=\langle-2,\infty)\), \(D_{f'}=(-2,\infty)\)

• Variant 5

  \( \arctan\frac1x \)

• Result

  \(\displaystyle f'=-\frac1{x^2+1}\), \(D_f=D_{f'}=\mathbb R\)

• Variant 6

  \( \arctan{\sqrt x}\)

• Result

  \(\displaystyle f'=-\frac1{2(x+1)\sqrt{x}} \), \(D_f=\langle0,\infty)\), \(D_{f'}=(0,\infty)\)

• Variant 7

  \(\displaystyle \arcsin\left(\frac{2x}{1+x^2}\right) \)

• Resolution

  \(\displaystyle \left(\arcsin\left(\frac{2x}{1+x^2}\right)\right)'= \frac1{\sqrt{1-\left(\frac{2x}{1+x^2}\right)^2}}\cdot\frac{2(1+x^2)-4x^2}{(1+x^2)^2}= 2\frac{1+x^2}{\sqrt{(1+x^2)^2-(2x)^2}}\cdot\frac{1-x^2}{(1+x^2)^2}= \frac{2}{1+x^2} \)

• Result

  \(\displaystyle f'= \frac{2}{1+x^2} \), \(D_f=D_{f'}=\mathbb R\)

• Variant 8

  \( \arctan(x^2) \)

• Result

  \(\displaystyle f'=\frac{2x}{x^4+1} \), \(D_f=D_{f'}=\mathbb R\)

• Variant 9

  \(\displaystyle \arcsin\sqrt{\frac{x-1}{x+1}}\)

• Resolution

  \(\displaystyle \left(\arcsin\sqrt{\frac{x-1}{x+1}}\right)'= \frac1{\sqrt{1-\frac{x-1}{x+1}}}\cdot\frac1{2\sqrt{\frac{x-1}{x+1}}}\cdot\frac{2}{(x+1)^2}=\frac{1}{\sqrt2(1+x)\sqrt{1-x}}\)

• Result

  \(\displaystyle f'= \frac{1}{\sqrt2(1+x)\sqrt{1-x}}\),

  \(D_f=(-\infty,-1)\cup\langle 1,\infty), D_{f'}=(-\infty,-1)\cup(1,\infty)\)

• Variant 10

  \(\displaystyle \arctan\left(\frac{x-1}{x+1}\right) \)

• Resolution

  \(\displaystyle \left(\arctan\left(\frac{x-1}{x+1}\right)\right)'= \frac1{1+\left(\frac{x-1}{x+1}\right)^2}\cdot\frac{2}{(x+1)^2}= \frac{1}{x^2+1} \)

• Result

  \(\displaystyle f'= \frac{1}{x^2+1} \), \(D_f=D_{f'}=\mathbb R\setminus\{-1\}\)

• Variant 11

  \( \ln(\cos x) \)

• Result

  \(\displaystyle f'=-\tan(x)\), \(D_f=D_{f'}=(-\frac\pi2,\frac\pi2)+2k\pi\)

• Variant 12

  \( \ln\left(\arctan \sqrt{x}\right) \)

• Result

  \(\displaystyle f'=\frac1{\arctan \sqrt{x}}\cdot\frac{1}{1+x}\cdot\frac{1}{2\sqrt x} \), \(D_f=D_{f'}=(0,\infty)\)