Answers: The chain rule
These are the answers to Questions: The chain rule.
Please attempt the questions before reading these answers!
1.1. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left( \dfrac{1}{7}\cos(5+4x)\right)=-\frac{4}{7}\sin(5+4x).\)
1.2. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(4\cos(x^2)\right)= -8x \sin(x^2).\)
1.3. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(e^{x^2 + 5}\right) = 2xe^{x^2+5}\)
1.4. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(2(\sin(2x))^2\right)= 8\sin(2x)\cos(2x).\)
1.5. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(e^{\sin(3x)}\right)=3\cos(3x)e^{\sin(3x)}.\)
1.6. Using laws of logarithms, write \(\ln((2+4x^{-2})^{-1})=-\ln(2+4x^{-2})\). Then \[\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\ln({{2+ 4x^{-2}})^{-1}}\right)=\frac{8x^{-3}}{2+4x^{-2}} =\frac{4}{x^3+2x}.\]
1.7. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(e^{5x^4}\right)= 20x^3e^{5x^4}\)
1.8. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(e^{2x^{-3}}\right)= -6x^{-4}e^{2x^{-3}}\)
1.9. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(-5\sqrt{x-2}\right)=-\frac{5}{2\sqrt{x-2}}\)
1.10. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\sqrt{(x+3)^2}\right)= 1\) (for \(x\geq 3\).)
1.11. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\ln (x^2 +1)\right)=\frac{2x}{x^2+1}\)
1.12. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\ln(\cos(x))\right)=-\tan(x)\).
1.13. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(2\cos^2(x)\right)=-4\cos(x)\sin(x)=-2\sin(2x).\)
1.14. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(2(x^3 + 5x^2 + 13x -1)^3\right)= 6(3x^2+10x+13)(x^3 + 5x^2 + 13x -1)^2.\)
1.15. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\sqrt{\dfrac{1}{2x}}\right)=-\frac{1}{4x^2}\left(\frac{1}{2x}\right)^{-1/2}\)
1.16. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\cos(5x^{-1/2})\right)=\frac{5}{2}x^{-3/2}\sin(5x^{-1/2}).\)
1.17. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\sin\left(\sqrt{x^2+1}\right)\right)= \frac{x\cos(\sqrt{x^2 +1})}{\sqrt{x^2+1}}\)
1.18. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\sin(e^x)\right)= e^x\cos(e^x).\)
1.19. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\cos(e^{-2x} + 5)\right)= 2e^{-2x}\sin(e^{-2x}+5).\)
1.20. \(\displaystyle \quad \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\ln(3x^3 + \sin(x))\right)= \frac{9x^2+\cos(x)}{3x^3+\sin(x)}.\)
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.