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

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Find derivatives using the chain rule I

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Q.

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

Identify function: Identify the function to differentiate. The function is $y=(1−x)^2(2x+3)$ .
Recognize product functions: Recognize that the function is a product of two functions, $u(x)=(1−x)^2$ and $v(x)=(2x+3)$ . We will need to use the product rule to differentiate $y$ .
Apply product rule: Apply the product rule. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. In formula terms, if $y = u(x)v(x)$ , then $y' = u'(x)v(x) + u(x)v'(x)$ .
Differentiate $u(x)$ : Differentiate the first function, $u(x)=(1−x)^2$ . Using the chain rule, the derivative of $u(x)$ is $2(1−x)(-1)$ , which simplifies to $-2(1−x)$ .
Differentiate $v(x)$ : Differentiate the second function, $v(x)=(2x+3)$ . The derivative of $v(x)$ is $2$ .
Apply product rule derivatives: Apply the product rule using the derivatives from steps $4$ and $5$ . So, $y' = u'(x)v(x) + u(x)v'(x) = -2(1−x)(2x+3) + (1−x)^2(2)$ .
Simplify derivative expression: Simplify the expression for the derivative. $y' = -4(1−x)(x+\frac{3}{2}) + 2(1−x)^2$ .
Expand and combine terms: Expand and combine like terms to get the final derivative. $y' = -4x - 6 + 4x^2 + 6x + 2 - 4x + 2x^2$ .
Combine like terms: Combine like terms to get the final simplified derivative. $y' = 6x^2 + 2x - 4$ .

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