How to apply the product rule in differentiation, Please see the attached image file for the question: I am not understanding the 3rd part of this solution.

Question

anonymous · Answer

attachment

phoenixfire · Answer

The third part just substitute values for u, dv/dx, v, and du/dx and then proceeded to expand and simplify.

anonymous · Answer

I am not getting this one: How did we get this? If simplification how can I do it. I am not good at simplification. 

$$(2x^2+6x)(6x^2+10x)+(2x^3+5x^2)(4x+6)$$

anonymous · Answer

How did we got this 2x^2 ?

phoenixfire · Answer

$$u=2x^2 + 6x$$
$$v=2x^3+5x^2$$
use differentiation rule to find the derivatives: $${dy \over dx}=ax^{a-1}$$
$${du \over dx}=4x+6$$
$${dv \over dx}=6x^2+10x$$
then substitute these values into the product rule equation: $${dy \over dx} = u{dv \over dx} + v{du \over dx}$$

anonymous · Answer

Thank you, I understood this part .. can you please tell me what next we have to do?

phoenixfire · Answer

After substituting you get this...
$$(2x^2+6x)(6x^2+10x) + (2x^3+5x^2)(4x+6)$$
then you expand the brackets and simplify. giving you:
$$20x^4+88x^3+90x^2$$

anonymous · Answer

In first part, it is cross multiplication, if I am right? In second part of simplification How did we got this 20x^4 ... and so on? its confusing

phoenixfire · Answer

(2x^2 * 6x^2) + (2x^2 * 10x) + (6x * 6x^2) + (6x * 10x) + (2x^3 * 4x) + (2x^3 * 6) + (5x^2 * 4x) + (5x^2 * 6)

(a + b)(c + d) = ac + ad + bc + bd

anonymous · Answer

ok got it .. and please tell me the last part :)

phoenixfire · Answer

well that was the expansion of brackets... now simplify the equation by combining like-terms. for example: 10x^2 + 3x^2 becomes 13x^2 because the x^2 is common.

anonymous · Answer

Sorry, still not getting it. I have took a lot of your time. If you can please provide me a link where I can understand the simplification process easily because I need to clear this concept of simplification for solving differentiation.

phoenixfire · Answer

well I'll do the first part, then you can do the second.
(2x^2+6x)(6x^2+10x)=(2x^2 * 6x^2) + (2x^2 * 10x) + (6x * 6x^2) + (6x * 10x)
do each term
2x^2 * 6x^2 = 12x^4    (ax^b * cx^d)=(ac)x^(b+d)
2x^2 * 10x = 20x^3
6x * 6x^2 = 36x^3
6x * 10x = 60x^2
put them all together
12^4 + 20x^3 + 36x^3 + 60x^2 
combine like-terms: 20x^3 + 36x^3 = 56x^3     ax^c + bx^c = (a+b)x^c
giving you 12x^4 + 56x^3 + 60x^2
now you go try the second half (the last four parts of the expansion)
then combine the terms with mine, and you should arrive at the answer