When you take the derivative of ##\arctan (y/x)## with respect to ##x##, you also get a ##-1## factor as ##x## is in the denominator. That cancels the first minus sign.influx said:
##x=r\cos \theta##, so ##u=U_\infty +\frac{m}{r}\cos \theta## becomes ##u=U_\infty+\frac{m}{x}## (the ##\cos \theta## cancels out, so it doesn't matter if it is +1 or -1).influx said:
I think it has to do with the way the arctan is used here. There are two arguments, x and y, instead of one. There is a special rule for the derivative of tan-1(y/x), as shown in this article:influx said:
The working out states that v = (m/r)sin(theta) but surely this should be (-m/r)sin(theta) ? I mean there is a negative in front of the ∂Ψ/dx ?
Also, if v=0 then 0 = (m/r)sin(theta), therefore sin(theta) = 0? And from this we know cos(theta) = 1 or -1 which would mean u = Uinfinity + (m/r) or Uinfinity - (m/r) as opposed to u = Uinfinity + (m/x) as they got?
Oops, the stated reason is wrong.Samy_A said:
A simple differentiation mistake is an error made during the process of finding the derivative of a function, typically in calculus. It can occur at any step of the process and can lead to incorrect results.
Some common examples of simple differentiation mistakes include forgetting to apply the chain rule, dropping a negative sign, or incorrectly applying the power rule.
To avoid making simple differentiation mistakes, it is important to carefully follow the steps of the differentiation process and double check your work. Practice and familiarity with the rules and concepts of differentiation can also help prevent mistakes.
The consequences of making a simple differentiation mistake can vary depending on the context. In some cases, it may lead to incorrect results and potentially affect the outcome of a problem or calculation. In other cases, it may simply require more time and effort to correct the mistake.
Yes, simple differentiation mistakes can be corrected by identifying the error and going back through the steps of the differentiation process to fix it. It is important to carefully review your work and check for any other potential mistakes before finalizing your solution.