1) y = (x² + 6)(x² - 3)^9 \frac{d}{dx}(x² + 6)(x² - 3)^9
1) y = (x² + 6)(x² - 3)^9 \frac{d}{dx}(x² + 6)(x² - 3)^9
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u=(x^{2}+6)) and (v=(x^{2}-3)^{9}). First, find (u^\prime) and (v^\prime). (u^\prime=\frac{d}{dx}(x^{2}+6) = 2x) For (v=(x^{2}-3)^{9}), use the chain rule. Let (t = x^{2}-3), then (v = t^{9}). By the chain rule (\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}). (\frac{dv}{dt}=9t^{8}) and (\frac{dt}{dx}=2x), so (v^\prime = 9(x^{2}-3)^{8}\cdot2x=18x(x^{2}-3)^{8})
Step2: Substitute into the product rule formula
(y^\prime=u^\prime v+uv^\prime) (y^\prime = 2x(x^{2}-3)^{9}+(x^{2}+6)\cdot18x(x^{2}-3)^{8})
Step3: Factor out common terms
Factor out (2x(x^{2}-3)^{8}) (y^\prime=2x(x^{2}-3)^{8}[(x^{2}-3)+9(x^{2}+6)]) Expand the brackets inside: ((x^{2}-3)+9(x^{2}+6)=x^{2}-3 + 9x^{2}+54) (=10x^{2}+51)
Answer:
(2x(x^{2}-3)^{8}(10x^{2}+51))