𝑦=(1−𝑥)^2(2𝑥+3)

Question

𝑦=(1−𝑥)^2(2𝑥+3)

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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+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.

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

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

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