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)(\square) + (x³ - 4x + 2)(\square)

Explanation:

Step1: Differentiate \(v = x^{3}-4x + 2\)

Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\), we have \(\frac{dv}{dx}=3x^{2}-4\)

Step2: Differentiate \(u = 3 - x^{2}\)

Using the power rule, \(\frac{du}{dx}=-2x\)

Step3: Apply the product rule \(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)

Substitute \(u = 3 - x^{2}\), \(\frac{dv}{dx}=3x^{2}-4\), \(v=x^{3}-4x + 2\) and \(\frac{du}{dx}=-2x\) into the product rule formula.
\(\frac{d}{dx}((3 - x^{2})(x^{3}-4x + 2))=(3 - x^{2})(3x^{2}-4)+(x^{3}-4x + 2)(-2x)\)

Answer:

\((3 - x^{2})(3x^{2}-4)+(x^{3}-4x + 2)(-2x)\)