differentiate the given function, simplify your answer by leaving no negative or rational exponents. final answer should be in factored form where possible.
y=(x^2 +2x)^3(x+2)
y = (x^2 +2x)^3(x+2) = x^3(x+2)^4
y' = 3x^2 (x+2)^4 + x^3 * 4(x+2)^3 * 1
= x^2(x+2)^3 (3(x+2) + 4x)
= x^2(x+2)^3(7x+6)
To differentiate the given function, we can use the product rule and the chain rule. Let's break it down step by step:
Step 1: Expand the expression (x^2 + 2x)^3
To expand (x^2 + 2x)^3, we can use the binomial theorem or simply multiply it out:
(x^2 + 2x)^3 = (x^2 + 2x)(x^2 + 2x)(x^2 + 2x)
= (x^4 + 4x^3 + 4x^2)(x^2 + 2x)
= x^6 + 4x^5 + 4x^4 + 2x^4 + 8x^3 + 8x^3
= x^6 + 4x^5 + 6x^4 + 16x^3
Step 2: Apply the product rule
Now that we have expanded the expression, we can differentiate it using the product rule. The product rule states that if you have two functions u(x) and v(x), then the derivative of their product (u(x) * v(x)) is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
Let's consider u(x) = (x^2 + 2x)^3 and v(x) = (x + 2).
Differentiating u(x):
We can use the chain rule here. So,
u'(x) = 3(x^2 + 2x)^2 * (2x + 2)
Differentiating v(x):
We differentiate v(x) directly, which gives us:
v'(x) = 1
Now we can apply the product rule:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
= 3(x^2 + 2x)^2 * (2x + 2) * (x + 2) + (x^6 + 4x^5 + 6x^4 + 16x^3) * 1
= 3(x^2 + 2x)^2 * 2(x + 2) * (x + 2) + (x^6 + 4x^5 + 6x^4 + 16x^3)
= 3(x^2 + 2x)^2 * 2(x + 2)^2 + (x^6 + 4x^5 + 6x^4 + 16x^3)
Step 3: Simplify the answer
To simplify the answer, we can multiply out the terms and combine like terms:
3(x^2 + 2x)^2 * 2(x + 2)^2 + (x^6 + 4x^5 + 6x^4 + 16x^3)
= 3(4x^4 + 8x^3 + 4x^2) * 2(x^2 + 4x + 4) + x^6 + 4x^5 + 6x^4 + 16x^3
= 24x^6 + 96x^5 + 144x^4 + 192x^3 + 72x^4 + 288x^3 + 432x^2 + x^6 + 4x^5 + 6x^4 + 16x^3
= 25x^6 + 100x^5 + 222x^4 + 496x^3 + 432x^2
This is the final answer.