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When we find the derivative of $z^2$ with respect to $z$ it means the slope of the graph,Which comes out to be $2z$. $$ \frac{dz^2}{dz}=2z $$if we take $dz$ on the other side it becomes $dz^2=2zdz$ which is known as the differnetial of $z^2$. I am not sure what this means, does it mean if we change $z^2 $with $dz$ then $z$ changes by $2zdz$?

If so, then for example I have an equation $z^2+\cos\theta+26$. Then can I differentiate the sides and get $$2zdz-\sin\theta d\theta=0$$ What does this even mean?

uniquesolution
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Naruto
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  • $z^2+\cos \theta + 26$ is not an equation... And if you differentiate $z^2+\cos \theta + 26=0$ with respect to $z$ you just get $2zdz = 0$, unless $\theta$ depends on $z$ for some reason. So what are you actually doing in the last step? – Gibbs Jun 04 '20 at 09:51

1 Answers1

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It means that if you change z by dz, then what is the corresponding change in $z^2$

To the 2nd part there should be a 0 on LHS otherwise it's not an equation. If there is a 0 on LHS then what you have done is right. It means to see how both the changes dz and $d\theta$, depend on each other, ie what is the change $d\theta$ in $\theta$ when z is changed by dz.

This dependence is implicitly contained in the eqn $z^2 + \cos\theta+26=0$

Shashaank
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  • If z=f(theta) and you differentiate both sides,I know what is the change in z2 when z is changed by dz but i dont know what is the change in costheta when z is changed by dz,The change i got fir costheta is when theta is changed by dtheta so are they not to separate things how can I equate them? – Naruto Jun 04 '20 at 11:03
  • I thin i understood what you mean.Tell me if i am correct – Naruto Jun 04 '20 at 11:06
  • As z=f(theta) when z is changed by dz theta is changed by theta so when z^2 is changed by dz^2 z is changed by 2zdz and so when z is changed theta will also change and so cos theta will also change bu not 26.But I had just one doubt z is changed by 2zdz but then theta cannot change by just dtheta as that is the change when z is changed by dz.So how does it become sintheta – Naruto Jun 04 '20 at 11:10
  • I don't get what you are asking. If want the change in z that is dz it is by simple substitution $ dz = \frac{\sin\theta d\theta}{2z}$ – Shashaank Jun 04 '20 at 11:27
  • @Naruto and z^2 is not changes by dz^2... Rather when z is changed by dz then z^2 is changes by 2zdz – Shashaank Jun 04 '20 at 11:28
  • Ohk thanks a lot – Naruto Jun 04 '20 at 11:38
  • But isnt dz^2=2zdz what does this mean? – Naruto Jun 04 '20 at 11:39
  • @Naruto if you mean d(z^2) = 2zdz you are correct .. If you mean dz^2 then I will read it as "dz squared" which is a second order change in z. I guess you meant 1st one so you are correct although since you are not using latex it becomes difficult to understand what you are trying to say.. – Shashaank Jun 04 '20 at 11:42