Question

really stuck on these two, steps would be greatly appreciated. gotta find the derivative thanks!

first one:

f(t)=2^(log5t)

the 5 is the base


second one:

y=(2(x^2) - 1)^5 /(√(x+1)


for the second one you have to use logarithmic differentiation

Answer 1

recall that (a^u)' = lna a^u u', so we have

f' = ln2 2^log5t (log5t)'

Now, log5t = lnt/ln5, so

f' = ln2 2^log5t * 1/ln5 * 1/t
f' = ln2/ln5 * 1/t 2^log5t

Now, ln2/ln5 = log5(2), so finally,

f' = (log5(2) / t) 2^log5t

y = u^5/v where
u = 2x^2-1
v = √(x+1)

y' = (5u^4 u' v - u^5 v')/v^2
= u^4 (5vu' - v')/v^2
= (2x^2-1)^4 (5(2x^2-1) - 1/(2√(x+1)))/(x+1)
You can massage this as you want; one form is
(2x^2-1) (38x^2+40x+1)
------------------------------
2(x+1)√(x+1)

Answer 2

oops. forgot an ^4 on the last line