Differentiating Complicated Functions

Need help solving some of these. We're reviewing for an exam.

First problem:

(x^1/2)+(x^-4/3)+(2^x)-(x^pi)+(e^2)

I got some of it but I'm stuck on the last three expressions.

Second one:

ln[(x^2)(2^x)]

I don't know where to start with derivatives that involve the ln. I think I missed that section entirely. If anyone has a link to resources that will help with ln derivatives, that'd be great too.

Third problem:

ln[csc(5x)-cot(5x)]

x^(1/2) + x^(-4/3) + 2^x − x^π + e^2

First two are simple enough. Just apply power rule:
d/dx (x^n) = n x^(n-1)
Note that 4th one also uses power rule, where n = π (since π is a constant exponent)

Derivative of e^2 is also very easy, since e^2 is a constant
and derivative of constant = 0

The more complicated one is 2^x
d/dx (e^x) = e^x
Note that number "a" can be written as a = e^ln(a)
d/dx (a^x) = d/dx ((e^lna)^x)
. . . . . . . . . = d/dx (e^(x lna))
. . . . . . . . . = e^(x lna) * d/dx (x lna)
. . . . . . . . . = e^(x lna) * lna
. . . . . . . . . = a^x * ln(a)

Therefore, d/dx (2^x) = 2^x ln(2)

d/dx (x^(1/2) + x^(−4/3) + 2^x − x^π + e^2)
= 1/2 x^(−1/2) − 4/3 x^(−7/3) + 2^x ln(2) − π x^(π−1) + 0

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Let y = ln(x)
e^y = x -------> differentiate both sides with respect to x
e^y dy/dx = 1
x dy/dx = 1
dy/dx = 1/x

d/dx (ln(x)) = 1/x
By chain rule: d/dx (ln(f(x)) = 1/f(x) * f'(x) = f'(x) / f(x)

d/dx (ln(x² 2^x))
= d/dx (ln(x²) + ln(2^x))
= d/dx (2 ln(x) + x ln(2))
= d/dx (2 ln(x)) + d/dx (x ln(2))
= 2 * 1/x + ln(2) * 1
= 2/x + ln(2)

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d/dx (ln(csc(5x) − cot(5x))
= 1/(csc(5x) − cot(5x)) * d/dx (csc(5x) − cot(5x))
= (−5 cot(5x)csc(5x) + 5csc²(5x)) / (csc(5x) − cot(5x))
= 5 csc(5x) (cos(5x) − cot(5x)) / (csc(5x) − cot(5x))