Differentiation??

Question

How do you differentiate using the product rule?

For example how do you differentiate y=x²(2x+5)³

Answer 1

let x² = u
let (2x+5)³ = v

product rule = uv' + vu'

so the solution is (2x)(2x+5)³ + ((x²)(3(2x+5)²*(2))

and u multiply and add

and this part 3(2x+5)²*(2) is dute to the differentiation of the 2x+5 which gives 2 which is why u multiply the differentiated function by two, if u still dont understand it email me

Answer 2

If you differentiate by the product rule
y' = x²[(2x+5)³]' + [x²]'(2x+5)³

Look at the first term:
To differentiate [(2x+5)³] use the chain rule:
z = [(2x+5)³]
Let u = 2x + 5, du = 2 dx
z = u^3
dz = 3u^2 du
Chain rule: (dz/dx) = (dz/du)(du/dx)
dz/dx = (3u^2) *(2), but u = 2x + 5
dz/dx = 6(2x + 5)^2

Now
y' = x^2* 6(2x + 5)^2 + [x²]'(2x+5)³
y' = 6x^2(2x + 5)^2 + [x²]'(2x+5)³
The second term is trivial to differentiate
y' = 6x^2(2x + 5)^2 + 2x(2x+5)³

Answer 3

dy / dx
= ( 2 x ) (2 x + 5) ³ + ( 3 )( 2 x + 5 ) ² (2 x) (x ²)
= (2 x ) (2 x + 5) ³ + ( 6 x ³ ) (2 x + 5) ²
= (2 x) (2 x + 5) ² (2x + 5 + 3 x ²)
= (2 x) (2 x + 5) ² (3 x ² + 2 x + 5)

Answer 4

u'v + uv' is the product rule.
Then use the chain rule for (2x+5)³.

y' = 2x(2x+5)³ + x² * 3(2x+5)² * 2 =

2x(2x+5)³ + 6x²(2x+5)²