SOLUTION: Could you please help me with this Calculus problem? I am finding this problem hard to solve. After a few steps, I am stuck and don't know how to proceed to find the derivative.

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Question 1203615: Could you please help me with this Calculus problem? I am finding this problem hard to solve. After a few steps, I am stuck and don't know how to proceed to find the derivative.
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Found 3 solutions by math_helper, ikleyn, greenestamps:
Answer by math_helper(2461)   (Show Source): You can put this solution on YOUR website!

h(x) =
The product rule can be applied:
If h(x) = A(x)B(x) then

h'(x) = A(x)B'(x) + B(x)A'(x)
Choosing A(x) = (3x-12) and B(x) =
A'(x) = 3
Unfortunately, B'(x) will be messy:
B'(x) =

Now you have:
h'(x) = +

Expand the first term, combine like-terms, and pull out , and you should get:
h'(x) =
[ Sometimes writing as x^(1/2) makes the workout easier. ]
---
EDIT 9/14/23: tutor greenestamps answer is missing a factor of

Answer by ikleyn(52784)   (Show Source): You can put this solution on YOUR website!
.

Formally speaking, to solve such problem successfully, you need to know pre-requisites.
The pre-requisites are:

    (a) knowing what a derivative is, in general;

    (b) knowing derivatives of table functions;

    (c) knowing the rule of taking derivative for a product of two functions

            ((f(x)*(g(x))' = f'(x)*g(x) + f(x)*g'(x)

        and for a product of three functions

            (f(x)*g(x)*h(x))' = f'(x)*g(x)*h(x) + f(x)*g'(x)*h(x) + f(x)*g(x)*h'(x).


Formally, that is all.


But, to be honest, in addition to it, you should have another pre-requisite,
which is the developed technique of solving more simple problems.


To develop such a technique, solve  a series of more simple problems,
listed below.  Take the derivatives of these functions

    (a)  ;

    (b)  ;

    (c)  .


As soon as you complete these problems, you will be ready to work on your
problem in the post.  Then it will be not so difficult exercise for you.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


A basic application of the product rule for derivatives tells us

(d/dx)((f(x)e^x)) = f(x)e^x+f'(x)e^x

So in this problem probably the easiest way to find h'(x) is to write f(x)=(3x-12)(x^4-x^(1/2)) and find the derivative using that pattern.

f(x) = (3x-12)(5x^4-x^(1/2)) = 15x^5-60x^4-3x^(3/2)+12x^(1/2)

f'(x) = 75x^4-240x^3-(9/2)x^(1/2)+6x^(-1/2)

f(x)+f'(x) = 15x^5+15x^4-240x^3-3x^(3/2)+(15/2)x^(1/2)+6x^(-1/2)

And so

ANSWER: h'(x) = (15x^5+15x^4-240x^3-3x^(3/2)+(15/2)x^(1/2)+6x^(-1/2))


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