Hyperbolic function and the product rule.

[titowakoru]

Homework Statement

The question I am trying to answer requires me to find the following:

dN/dS ∝ S^−5/2/cosh(r/R)

and I am giving the follwing equation in the question.

A=4πR^2 sinh^2⁡〖(r/R)〗

The Attempt at a Solution

Right I know how to get the S^-5/2 in the top half of the equation.

I also understand that the cosh part comes from the differentiation of A. The problem I have is after applying the product rule to A I end up with this:

dA/dr=8πRsinh^2 (r/R)+ 8πR^2 sinh⁡(r/R)cosh⁡(r/R)

I am stuck on how the terms cancel to leave me with cosh(r/R) so i can reach the equation required.

 

[Mark44]

Homework Statement

The question I am trying to answer requires me to find the following:

dN/dS ∝ S^−5/2/cosh(r/R)

and I am giving the follwing equation in the question.

A=4πR^2 sinh^2⁡〖(r/R)〗

The Attempt at a Solution

Right I know how to get the S^-5/2 in the top half of the equation.

I also understand that the cosh part comes from the differentiation of A. The problem I have is after applying the product rule to A I end up with this:

dA/dr=8πRsinh^2 (r/R)+ 8πR^2 sinh⁡(r/R)cosh⁡(r/R)

I am stuck on how the terms cancel to leave me with cosh(r/R) so i can reach the equation required.

The product rule doesn't apply here, but the chain rule does. The factor 4πR² is considered to be a constant as far as differentiation with respect to r is concerned.

 

[titowakoru]

Ah, I see. So applying the chain rule with respect to r would mean that 4πR^2 would disappear because the derivative of a constant is zero?

 

[SteamKing]

If y = c * f(x), then dy/dx = c * d[f(x)]/dx