Collection of Maths Problems

Derivatives of ratios, roots and logarithms

Task number: 3010

• Variant 1

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

• Result

  \(\displaystyle f'=\frac2{(x+1)^2} \), \(D_f=D_{f'}=\mathbb R\setminus\{-1\}\)

• Variant 2

  \(\displaystyle \frac2{(x^2-1)^3} \)

• Result

  \(\displaystyle f'=-\frac{12x}{(x^2-1)^4} \), \(D_f=D_{f'}=\mathbb R\setminus\{-1{,}1\}\)

• Variant 3

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

• Result

  \(\displaystyle f'=\frac{4(x-1)}{(x+1)^3} \), \(D_f=D_{f'}=\mathbb R\setminus\{-1\}\)

• Variant 4

  \(\displaystyle \sqrt{\frac{x+2}{x-1}} \)

• Result

  \(\displaystyle f'=\frac{3}{2\sqrt{(x+2)(x-1)^3}} \), \(D_f=(\infty,-2\rangle\cup(1,\infty), D_{f'}=(\infty,-2)\cup(1,\infty)\)

• Variant 5

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

• Result

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

• Variant 6

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

• Result

  \(\displaystyle f'=-\frac1{2x\sqrt{1-\ln x}} \), \(D_f=(0,e\rangle, D_{f'}=(0, e)\)

• Variant 7

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

• Result

  \(\displaystyle f'=\frac2{x^2+1}\), \(D_f=D_{f'}=(\infty,-1)\cup(1,\infty)\)

• Variant 8

  \( \ln\left(x+\sqrt{1+x^2}\right) \)

• Result

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

• Variant 9

  \( x^2\sqrt{\ln x-1} \)

• Result

  \( \displaystyle f'=2x\sqrt{\ln x-1}+\frac{x}{2\sqrt{\ln x-1}} \), \(D_f=D_{f'}=(e,\infty)\)

• Variant 10

  \( \ln(\ln(\ln x)) \)

• Result

  \( \displaystyle f'=\frac1{x\ln x \ln(\ln x )} \), \(D_f=D_{f'}=(e,\infty)\)

• Variant 11

  \( \ln\left(1+e^{-x^2}\right) \)

• Result

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