# derivative calculus

f(t)=(4+√t)(4t^2-5)

This is what I have so far...

(4+√t)(8t)+(4t^2-5) (1/2t^-1/2)

Am struggling to get this in factored form. Am working on a multiple choice take home test and nothing I am getting matches.

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1. the derivative you got is correct.
by distributing, we can further simplify this into:
32t + 8t^(3/2) + 2t^(3/2) - (5/2)t^(-1/2)
10t^(3/2) + 32t - (5/2)t^(-1/2)

hope this helps~ :)

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2. sofar ok
why don't we expand this ... and simplify to get

32t + 10t^(3/2) - (5/2)t^(-1/2)
= 32t + 10t√t - 5/(2√t)

Since I don't know what your choices are, I don't know in which direction to go from here.

Here is a good trick to see if your answer is the same as theirs.
Pick any point on the original
e.g. let t = 1
f(1) = 5(4-5) = -5

so (1,-5)
so sub in t = 1 into our derivative answer to get
32(1) + 10(1)(1) - 5/(2(1)) = 39.5
Any of the choices not giving you 39.5 would be incorrect.

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3. Here are my options

a) 10t^2+32t√t - 5 / √t

b) 20t^2-64t√t - 5 / 2√t

c) 20t^2+ 32t√t - 5 / 2√t

d) 20t^2+64t√t - 5 / 2√t

e) 10t^2-32t√t - 5 / √t

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5. If c) is written as
(20t^2 + 32t√t - 5)/(2√t) it would be same as my answer.
The formed a common denominator

the way you typed it, only the -5 would be divided by 2√t

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6. Thanks. I see my mistake now.

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