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What is the derivative of the pi function
I like Serena said:This is not quite right.
The proper derivative is:
[tex]\frac{d}{dn}\pi(n)\approx \frac{d}{dn}\int_{2}^{n}\frac{dt}{\ln(t)} = \frac{1}{\ln(n)}[/tex]
dimension10 said:If you are talking about the prime number thing, I am not sure about the exact one but here is an approximation:
[tex]\pi(n)\approx \int_{2}^{n}\frac{dt}{\mbox{ln}(t)}[/tex]
[tex]\frac{d}{dn}\pi(n)\approx \frac{d}{dn}\int_{2}^{n}\frac{dt}{\mbox{ln}(t)}[/tex]
[tex]\frac{d}{dn}\pi(n)\approx \frac{d}{dn}\lim_{\delta t \rightarrow 0}\sum_{t=2}^{n}\frac{\delta t}{\mbox{ln}(t)}[/tex]
As the derivative of a sum is the sum of the derivatives,
[tex]\frac{d}{dn}\pi(n)\approx \lim_{\delta t \rightarrow 0}\sum_{t=2}^{n}\frac{d}{dn}\frac{\delta t}{\mbox{ln}(t)}[/tex]
[tex]\frac{d}{dn}\pi(n)\approx \lim_{\delta t \rightarrow 0}\sum_{t=2}^{n}-\frac{\delta t}{n \; {\mbox{ln}}^{2}(n)}[/tex]
[tex]\frac{d}{dn}\pi(n)\approx - \int_{2}^{n}\frac{dt}{n \; {\mbox{ln}}^{2}(n)} [/tex]
So that is the approximate rate of change of the pi function of t as t changes.
dimension10 said:Then where did I make a mistake?
dimension10 said:According to Wolfram Alpha,
[tex] \frac{d}{dn}(\int_{2}^{n}\frac{dt}{\ln(t)})=\frac{-n+n\; \mbox{ln}(n)+2}{n \;{\mbox{ln}}^{2}(n)}[/tex]
dimension10 said:I guess all three solutions are equal to each other and thus, correct?
I like Serena said:As MathematicalPhysicist already said, the first mistake is when you moved d/dn to the other side of the summation symbol.
This is not allowed, because the summation is dependent on n.
I like Serena said:You made another mistake when you differentiated the expression dependent on t with respect to n.
Since the expression is not dependent on n, the result is zero.
I think I know what happened. It must have again considered d as constant rather than an infinitesimal.There seems to be a simpler solution using the second fundamental theorem of calculus and that would yieldI like Serena said:How did you get WolframAlpha to say that?
I do not get that.
The derivative of the pi function is equal to zero, as it is a constant value and does not change with respect to any variable.
No, the pi function cannot be differentiated as it is a constant value and does not have a variable to differentiate with respect to.
The derivative of the pi function is significant in calculus and mathematics as it represents the rate of change of a constant value. It also helps in solving problems related to motion and optimization.
The derivative of the pi function is related to the area of a circle through the formula for finding the area of a circle: A = πr². The derivative of this formula is 2πr, which is the circumference of the circle. This shows that the derivative of the pi function is the rate of change of the area of a circle with respect to its radius.
Yes, the derivative of the pi function is used in real-life applications such as engineering, physics, and economics. It helps in solving problems related to optimization, motion, and rate of change.