Answers: The quotient rule
These are the answers to Questions: The quotient rule.
Please attempt the questions before reading these answers!
1.1. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{e^x}{x}\right)=\frac{e^x(x-1)}{x^2}.\)
1.2. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{e^{7x}}{x^5}\right)=\frac{(7x -5)e^{7x}}{x^{6}}.\)
1.3. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\ln(x)}{x^2}\right)=\frac{1-2\ln(x)}{x^3}\)
1.4. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{e^{-x}}{x^2+11x-2}\right)= -\frac{(x^2 +13x +9)e^{-x}}{(x^2 + 11x -2)^2}\)
1.5. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{x^3+5x-5}{x^2+3}\right)=\frac{(3x^2+5)(x^2+3)-(x^3+5x-5)(2x)}{(x^2+3)^2}.\)
1.6. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\cos(x)}{x^2+3x-1}\right)=\frac{-\sin (x)(x^2+3x-1)-\cos(x)(2x+3)}{(x^2+3x-1)^2}.\)
1.7. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\tan(x)}{\cos(x)}\right)=\frac{\sec^2(x)\cos(x) + \tan(x)\sin(x)}{\cos^2(x)}\)
1.8. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\ln(3x)}{\ln(5)+x}\right)= \frac{1}{x(\ln(5)+ x)} - \frac{\ln(3x)}{(\ln(5) + x)^2}\)
1.9. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{x^2+3x}{\cos(x)}\right)=\frac{(2x+3)\cos(x) + (x^2+3x)\sin(x)}{\cos^2(x)}\)
1.10. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\ln(x)}{x^3+3}\right)=\frac{(x^3+3)-3x^3\ln (x)}{x(x^3+3)^2}\)
1.11. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{5\tan(x)}{x}\right)=\frac{5x\sec^2(x)-5\tan (x)}{x^2}.\)
1.12. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{3x^7-27x^2+2\sqrt{x}}{x^2+1}\right)=\frac{15x^8 \sqrt{x} +21x^6\sqrt{x} - 54x\sqrt{x} - 4x^2 + x^2+1}{\sqrt{x}(x^2+1)^2}.\).
1.13. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{e^{-3x}}{e^{2x}}\right)=\frac{-3e^{-3x}\,e^{2x}-2e^{-3x}\,e^{2x}}{e^{4x}}=-5e^{-5x}.\)
1.14. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{e^3x^3}{e^x}\right)= \frac{3e^{3+ x}x^2-e^{3+x}x^3e^x}{e^{2x}}= \frac{x^2e^{x+3}(3+ x)}{e^{2x}}.\)
1.15. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{x^5}{x^5+1}\right)=\frac{5x^4(x^5+1)-x^5(5x^4)}{(x^5+1)^2} = \frac{x^4}{(x^5+1)^2}\)
1.16. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\tan(x)}{\ln(x)}\right)=\frac{x\sec^2(x)\ln (x)-\tan(x)}{x(\ln(x))^2}.\)
1.17. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{3\sin(x)}{\ln(x)}\right)=\frac{3x\cos(x)\ln (x)-3\sin(x)}{x(\ln(x))^2}\)
1.18. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\tan(x)+5x}{\sec(3x)}\right)= \frac{\sec^2(x) +5 - (3\tan(3x))(5x + \tan(x))}{\sec(3x)}.\)
Version history and licensing
v1.0: initial version created 05/25 by Sara Delgado Garcia as part of a University of St Andrews VIP project.