Calculus - The sum rule

Question

I have the following example but can’t understand why the constants k, T and Eg haven’t been differentiated in the solution.

Question:

The probability for an atomic nucleus with energy E in the core of a star to undergo nuclear fusion depends on two terms:

p(E) =-(E/kT) - (Eg/E)^1/2, where k, T and Eg are constants. Find dp/dE

Solution:

dp/dE = (Eg/4E^3)^1/2 -1/kT

Answer 1

The reason is that they are constants. One of the fundamental rules of differentiation is that the derivative of a function times a constant is the same as the constant times the derivative of the function. Written out as an equation, this principle is:

d/dx(k*f(x)) = k*d/dx(f(x))

In other words the constants can be factored out of the differentiation. Here in an example using the sum rule, you must take the derivative of each term separately if you want to factor out constants to make the differentiation more visually simple.

p(E)=(-1/kT)E - (Eg)^1/2(E)^-1/2 Now the constants have been factored out of each term, we can apply the sum rule for differentiation:

dp/dE=(-1/kT)*d/dE(E) - (Eg)^1/2*d/dE(E^-1/2)

dp/dE=(-1/kT)*1 - (Eg)^1/2*(-1/2)*E^-3/2

and now just simplify:

dp/dE=(Eg/4E³)^1/2-1/kT