Please help with differentiation

Problem -

Find all values of x=c so that the tangent line to the graph of

f(x) = 4x^3(14x^2 +7x-21)^2 at (c,f(c)) will be horizontal.

the the solution the book gives me is 4x^3(14x^2+7x-21)(28x+7) + (12x^2)(14x^2+7x-21)^2 - this part i know how to get to. then 196x^2(x-1)(14x-9)(3+2x)(x+1) - which i don't know how they got to.

can someone please help explain how the book get the last part of the solution? thank you!

Re: Please help with differentiation

Quote:

Originally Posted by

**arcticreaver** Problem -

f(x) = 4x^3(14x^2 +7x-21)^2 at (c,f(c)) will be horizontal.

...

the the solution the book gives me is 4x^3(14x^2+7x-21)(28x+7) + (12x^2)(14x^2+7x-21)^2 - this part i know how to get to. then 196x^2(x-1)(14x-9)(3+2x)(x+1) - which i don't know how they got to.

The 1st term of what you quoted as the book's derivative is missing a factor of 2 from the differentiation of (14x^2 +7x-21)^2. Perhaps they corrected that error in a later step?

It should read:

$\displaystyle \frac{d}{dx}\left(4x^3(14x^2 +7x-21)^2\right) = $

$\displaystyle \left(4x^3\right) \left( 2(14x^2+7x-21)^1(28x+7) \right) + \left( 12x^2 \right)(14x^2+7x-21)^2 \right)$.

By the way, if you intend on working that out, I'd strongly suggest getting those ugly factors of 7 factored out before you even begin:

$\displaystyle 4x^3(14x^2 +7x-21)^2$

$\displaystyle = 4x^3 \left[ (7) (2x^2+x-3) \right]^2 = 4x^3 (7)^2 (2x^2+x-3)^2 = cx^3(2x^2+x-3)^2$, where c = (4)(49) = 196.

One more hint - when computing a derivative of a polynomial that you'll eventually want to factor, don't expand the polynomial. Keep it together.

What I'm saying here is do not try to expand $\displaystyle (14x^2 +7x-21)^2$ into a 5 term 4th degree polynomial in x. Keep it factored.

One final hint: You can factor $\displaystyle 2x^2+x-3$

Re: Please help with differentiation

2x^2+x-3 factored would be (2x+3)(x-1) and i see that this is part of the solution. but what happens to the (12x^2) and the (28x+7)?

Re: Please help with differentiation

1. Get rid of (i.e. factor out) those horrible 7's as I showed before.

2. Take the derivative. Note that because you've used the product rule, you'll have two terms, each with several factors.

3. Look for common factors between those two big terms. There will be some. Factor them out.

4. After #3, look at the "stuff inside" that remains after factoring out all the common terms. Simplify it (expand out and combine like terms, etc), and then *factor* the result.

If you do 1-4 correctly, you'll be looking at the correct factorization of the derivative - and from there, if you conceptually understand what's going on, you can pretty much write down the answer to the problem.