calculus

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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

  • calculus -

    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)

  • calculus -

    oops. forgot an ^4 on the last line

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