Questions: The product rule
Before attempting these questions, it is highly recommended that you read Guide: The product rule..
In this question set, the following definitions are used:
\[\cosh(x) = \frac{e^{x} + e^{-x}}{2} \quad\textsf{ and }\quad \sinh(x) = \frac{e^{x} - e^{-x}}{2}\] These are hyperbolic trigonometric functions; for more about these, see [Guide: Introduction to hyperbolic functions].
Differentiate the following functions using the product rule.
1.1. \(\quad xe^x\)
1.2. \(\quad x^2e^{2x}\)
1.3. \(\quad 5x^3\tan(x)\cos(x)\)
1.4. \(\quad x \ln(x)\)
1.5. \(\quad (x^3 + x^2 -5)(x +1)\)
1.6. \(\quad (13x^2 + 5x +2)(x^3 + 2)\)
1.7. \(\quad x(5x^2 + 3x + 2)(x^2 +x +1)\)
1.8. \(\quad (10x^2 + 21)\cos(x)\)
1.9. \(\quad \cosh(2x)\sinh (3x)\)
1.10 \(\quad (x^2 +3)\ln(x)\)
1.11 \(\quad \sin(x)\sqrt{x}\)
1.12 \(\quad \cosh(x)\ln(x)\)
1.13. \(\quad x^2\sqrt{x}+x^2\cos(x)\)
1.14. \(\quad e^{-5x}(x^3+5)\)
1.15. \(\quad \cos(x)\ln(x)\)
1.16. \(\quad \ln(x) \ln (3x) \ln (100x)\)
1.17. \(\quad (x^2 + 5x + 2)\sin(x)\)
1.18. \(\quad -\ln(x)\ln(3x)\)
1.19. \(\quad (x^5 + 3)(x^2+ 3x)(x^7 + x^4)\)
1.20. \(\quad (\sin(x) + 3x)e^{-x}\)
After attempting the questions above, please click this link to find the answers.
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.