find ( y ) by (a) applying the product rule and (b) multiplying the factors to produce a sum of simpler…

find ( y ) by (a) applying the product rule and (b) multiplying the factors to produce a sum of simpler terms to differentiate.\n( y=(3 - x^{2})(x^{3}-4x + 4) )\n\n a. apply the product rule. let ( u=(3 - x^{2}) ) and ( v=(x^{3}-4x + 4) ).\n( \frac{d}{dx}(uv)=(3 - x^{2})(square)+(x^{3}-4x + 4)(square) )

find ( y ) by (a) applying the product rule and (b) multiplying the factors to produce a sum of simpler terms to differentiate.\n( y=(3 - x^{2})(x^{3}-4x + 4) )\n\n a. apply the product rule. let ( u=(3 - x^{2}) ) and ( v=(x^{3}-4x + 4) ).\n( \frac{d}{dx}(uv)=(3 - x^{2})(square)+(x^{3}-4x + 4)(square) )

Answer

Explanation:

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

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{d}{dx}(x^{n})=nx^{n - 1}), we have (\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 + 4) and (\frac{du}{dx}=-2x) into the product rule formula. (\frac{d}{dx}(uv)=(3 - x^{2})(3x^{2}-4)+(x^{3}-4x + 4)(-2x))

Answer:

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