product rule to solve question derivative?

Question

How would you use the product rule to solve this question :

y (x) = (x^2+1)(x^3+1)

is this the next step: (2x)(x^3+1) + (3x2)(x^2+1) ??
I get confused with the algebra after that

the answer is y ' (x) = 5x^4+3^2+2x

but I would like someone to show me how step by step the algebra and did I set it up correctly?

Answer 1

Y(x) = (x^2+1)(x^3+1)
=d/dx/ (x^2+1) (x^3+1) + d/dx (x^3+1) (x^2+1)
= (2x)(x^3+1) + (3x^2)(x^2+1)
= (2x^4 + 2x) + (3x^4 + 3x^2)
= 5x^4 + 3x^2 + 2x

Answer 2

dy/dx = 2x(x^3+1) + (3x^2)(x^2+1)
= 2 x^4 + 2x + 3 x^4 + 3 x^2
= 5 x^4 + 3 x^2 + 2x

There is nothing wrong at all.

Answer 3

you've done the calculus correctly, to do the algebra just expand out both the left and right side of the + and then combine like terms:

(2x)(x^3+1) + (3x^2)(x^2+1)

2x^4 + 2x + 3x^4 + 3x^2

5x^4 + 3x^2 + 2x

Answer 4

first off in the next step it should read ...+ (3x^2)(x^2 +1)
after that you just multiply the numbers together bringing you to
2x^4 + 2x + 3x^4 + 3x^2
after that you add like terms and that will bring you to the answer
5x^4 + 3x^2 + 2x

hope this was helpful

Answer 5

Your first line is correct, from there you multiply out

2x^4 + 2x + 3x^4 + 3x^2 Then add like terms
2x^4 + 3x^4 + 3x^2 + 2x
5x^4 + 3x^2 + 2x

Answer 6

you're almost there
you might have gotten confused multiplying the exponents.

the next step is to just multiply it out

2x * x^3 = 2x^4
3x^2 * x^2 = 3x^4

so you multiply it all out and you end up with:
2x^4+ 2x+ 3x^4+ 3x^2

add:
5x^4 + 3x^2 + 2x

Answer 7

product rule

if y(x) = f(x)*g(x)

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

so if f(x) = x^2 + 1 then f'(x) = 2x
g(x) = x^3 + 1 then g'(x) = 3x^2

so y'(x) = 2x*(x^3+1) + (x^2+1)*3x^2

y'(x) = 5x^4 + 3x^2 + 2x

Answer 8

y = (x^2+1)(x^3+1)

let u = (x^2 +1), du = 2x
let v = (x^3 + 1), dv = 3x^2

y' = v*du + u*dv
= (x^3 + 1)*(2x) + (x^2 +1)*(3x^2)
= (2x^4 + 2x) + (3x^4 +3x^2)
= 5x^4 + 3x^2 + 2x

Answer 9

y `(x) = (2x) (x³ + 1) + (3x²).(x² + 1)
y `(x) = 2x^4 + 2x + 3x^4 + 3x²
y `(x) = 5 x^4 + 3x² + 2x

Answer 10

Well, this general method for all differentiation questions will clear your doubt.

f'(x) and g'(x) are derivatives of functions f(x) and g(x), respectively. f(x) and g(x) are two different functions (algebric expressions). f(x) can be equal to g(x), but then you won't have to follow the product rule. Simple differentiation will be all.

Let y = f(x) g(x).............(1)

Therefore, dy/dx = f(x) g'(x) + g(x) f'(x)

This is true for your question as well as many other questions that may have the expression (1).

And, yes your next step is correct.