calculus
posted by nicole on .
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

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^21
v = √(x+1)
y' = (5u^4 u' v  u^5 v')/v^2
= u^4 (5vu'  v')/v^2
= (2x^21)^4 (5(2x^21)  1/(2√(x+1)))/(x+1)
You can massage this as you want; one form is
(2x^21) (38x^2+40x+1)

2(x+1)√(x+1) 
oops. forgot an ^4 on the last line