Question

find y by (a) applying the product rule and (b) multiplying the factors to produce a sum of simpler terms to differentiate.
y = (3 - x²)(x³ - 4x + 2)
a. apply the product rule. let u = (3 - x²) and v = (x³ - 4x + 2).
\\(\frac{d}{dx}(uv) = (3 - x²)(3x² - 4) + (x³ - 4x + 2)(-2x)\\)
b. multiply the factors of the original expression, u and v, to produce a sum of simpler terms.
y = \\(\square\\)
(simplify your answer.)

Explanation:

Step1: Multiply the terms

$$\begin{align*} y&=(3 - x^{2})(x^{3}-4x + 2)\\ &=3\times(x^{3}-4x + 2)-x^{2}\times(x^{3}-4x + 2)\\ &=3x^{3}-12x+6-(x^{5}-4x^{3}+2x^{2}) \end{align*}$$

Step2: Simplify the expression

$$\begin{align*} y&=3x^{3}-12x + 6-x^{5}+4x^{3}-2x^{2}\\ &=-x^{5}+(3x^{3}+4x^{3})-2x^{2}-12x + 6\\ &=-x^{5}+7x^{3}-2x^{2}-12x + 6 \end{align*}$$

Answer:

\(y=-x^{5}+7x^{3}-2x^{2}-12x + 6\)