Differentiate and simplify completely.

[x^2(1-x)^3] / [(1+x)^3]

To differentiate and simplify the given expression, we'll follow these steps:

Step 1: Expand and simplify the numerator
Step 2: Simplify the denominator
Step 3: Divide the numerator by the denominator
Step 4: Differentiate the resulting expression
Step 5: Simplify the differentiated expression

Let's go through each step in detail:

Step 1: Expand and simplify the numerator
The numerator can be simplified by applying the binomial expansion to the term (1 - x)^3. The expansion is given by: (1 - x)^3 = 1^3 - 3 * 1^2 * x + 3 * 1 * x^2 - x^3
Simplifying this further, we get: (1 - x)^3 = 1 - 3x + 3x^2 - x^3

Now, we can multiply the numerator by x^2 and simplify further:
Numerator = x^2(1 - x)^3 = x^2(1 - 3x + 3x^2 - x^3)
= x^2 - 3x^3 + 3x^4 - x^5

Step 2: Simplify the denominator
The denominator, (1 + x)^3, is already simplified and does not require further simplification.

Step 3: Divide the numerator by the denominator
To divide the numerator by the denominator, we can write the given expression as:
Expression = (x^2 - 3x^3 + 3x^4 - x^5) / (1 + x)^3

Step 4: Differentiate the resulting expression
Differentiating the above expression requires the application of quotient rule, which states:
d/dx (u/v) = (v * du/dx - u * dv/dx) / v^2

Let's apply the quotient rule to differentiate the expression:
Expression' = [(1 + x)^3 * (2x - 9x^2 + 12x^3 - 5x^4) - (x^2 - 3x^3 + 3x^4 - x^5) * 3(1 + x)^2] / (1 + x)^6

Simplifying the numerator, we get:
Expression' = [(2x - 9x^2 + 12x^3 - 5x^4)(1 + x)^3 - 3(x^2 - 3x^3 + 3x^4 - x^5)(1 + x)^2] / (1 + x)^6

Step 5: Simplify the differentiated expression
Finally, simplify the resulting expression by expanding and collecting like terms. This will give you the completely simplified expression after differentiation.