# dependency

1. ### Proof of e^x, lnx and e as limit without circular dependency?

People prove the derivative of e^x, \ln(x) using either the formula of e = \lim_{x \to 0} (1+x) ^ \frac{1}{x} or if they know one of the derivatives e^x, \ln(x) they use it to prove the other. My textbook first gave me without proof that \frac{d}{dx}e^x = e^x Then it found the limit of \ln(x)...
2. ### Linear dependency and independence

Can this be explained by use of one solution, no solution or infinitely many solutions? If not, just summarize it for me the way you understand it. Sent from my SM-G955W using Tapatalk
3. ### Integral interval dependency

Physics student here, I'm doing some freelance analysis on planet formation and whatnot. When deriving equations to explain gravitational shrinking of a big body approximated to spheres, I got some expressions that would tell me the pressure, P, in function of a spherical radius, r. All well and...
4. ### Singular Matrix and Linear Dependency

I am asked to prove this: Matrix is singular<=>columns are linearly dependent<=>determinant is zero<=>matrix is singular I have proved everything except for the first implication, which I can't find anywhere. Can someone prove that if a matrix is singular, its columns are linearly dependent...
5. ### Relationship between singular matrices and linear dependency?

I know that when a matrix is singular it has no inverse and its determinant is 0. How does that relate to its linear dependency? How can I tell or prove this to myself? and is it the columns or rows that are linearly dependent?