Differentiate the following functions. You do not need to simplify.

Question

First question: f(x)=tan(lnx)+ln(5x^3/2-1)

For the first part, I got 1/x(sec²(lnx)) by doing the chain rule. The derivative of tan is sec²(x) and the derivative of ln(x) was 1/x, but I'm not sure how to do the second portion.

Second question: s(t)=e^3t+2+e^t*ln(2)

I changed the second portion e^t*ln2 to 2^t but didn't know how to proceed.

Third question: g(x)=cosx^cotx

Should I take the natural log of the equation?

Answer 1

f(x) = tan(lnx) + ln(5x^3/2 - 1)

f'(x) = sec²(lnx)(1/x) + (7.5√x) / (5x^3/2 - 1)

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s(t) = e^3t+2 + e^tln2 = e^3t+2 + e^{ln(2^t)} = e^3t+2 + 2^t

s'(t) = 3e^3t+2 + (ln2)2^t

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g(x) = cosx^cotx

Let y = cosx^cotx

Then lny = cotxln(cosx)

y'/y = -csc²x ln(cosx) + cotx(-sinx/cosx) = -csc²x ln(cosx) - 1

So, g'(x) = y' = cosx^cotx[-csc²x ln(cosx) - 1]