State the ideal gas law along with 3 assumptions that are made for the model to work
Ideal gas law: pV=nRT ,where p= pressure, V= volume, n= moles, R= gas constant (8.314) T= temperature. There are many assumptions made but the 3 simplest and most important are; the motion of the molecules is random, there are no intermolecular forces between them, and the molecules are assumed to be a point-like particle (i.e. they have negligible volume).
Use the product rule to differentiate (5x^2)(sin(x))
The product rule states that you must d/dx the first term then multiply it by the second then d/dx the second term and multiply it by the first, then add them together. In this case; d/dx of 5x^2 = 10x, so the first part of the addition is 10x*sin(x). The second part is (5x^2)*(d/dx sin(x)), which is 5x^2 * cos(x), so the final answer is 10x*sin(x) + 5x^2 * cos(x).
Physics is essential for describing the entire universe, from colliding supernovae to quantum entanglement, what is the approximate difference, given by 10^x, is the size of a carbon atom to the size of the observable universe?
The size of the observable universe is about 10^26 - 10^27 meters wide, whilst the diameter of a carbon atom is around 1 Angstrom, which is 10^-10m. So, the difference in orders of magnitude is 10^27 meters.