Recent content by S.R

The way I learned the chain rule (in the context of multivariable functions) was to draw a dependency diagram. In this case, however, the dependency diagram is not clear.

How would I obtain g_y though? I'm also not sure how to derive the first formula.

Should it be d/dx(siny) as in the single var. case?

Ah, thank-you. The first term cancels, right?

Homework Statement Let g(x, y) = f(sin(y), cos(x)). Find the second partial derivative of g with respect to x (g_xx). Homework EquationsThe Attempt at a Solution I attempted to find g_x, but I'm not entirely sure how chain rule applies in this situation. Is this correct? g_x = f_x(sin(y)...

Assuming x, y ∈ N: If x | y and x > y, then there exists an integer k such that kx = y or k = y/x. However, since x > y, the expression y/x is not an integer. Therefore, we can conclude x does not divide y, since no integer k exists such that kx = y.

I suppose if x and y are negative, then the converse is true. For instance, -2 | -4 is true, but -4 | -2 is false.

Question: If x | y, (is true), then x ≤ y and x ≠ 0. For instance, if x > y, then there are no integer solutions to equation kx = y and thus, x does not divide y. Is this a correct proposition?

Thank-you for the help!

(2n)^2 = 4n^2 (4n^2) mod 4 = 0 (assuming n is an integer). (2n+1)^2 = 4n^2 + 4n + 1. (4n^2 + 4n + 1) mod 4 = 1 (assuming n is an integer). Not particularly sure how this fact helps, though? EDIT: Oh, if we assume k is a perfect square, then k mod 4 = 0 or k mod 4 = 1 (depending on whether k...

I'm not sure what you mean by "perfect squares only take a few values modulo 8"?

I attempted this approach, but I guess I'll try again here. (x+a)^2 - x^2 = 10 x^2 + (2a)x + a^2 - x^2 = 10 (2a)x + a^2 = 10 a(2x + a) = 10 From here, I can presumably set a equal to factors of 10, right?

Ah, ok. Thank-you.

Homework Statement :[/B] Determine whether there exists an integer x such that x^2 + 10 is a perfect square. Homework Equations :[/B] N/A The Attempt at a Solution :[/B] Assume x^2 + 10 = k^2 (a perfect square). Solve for x in terms of k: x = ±sqrt(k^2 - 10) Since k is an integer and k^2 -...

Thanks for the response. I noticed the implementation of log laws in WolframAlpha's solution where log(a,x^2) was rewritten ln(x^2)/lna.