differentiate.\ny = 9x²(x - 5)⁵\ny =
differentiate.\ny = 9x²(x - 5)⁵\ny =
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = u\cdot v), then (y'=u'v + uv'). Let (u = 9x^{2}) and (v=(x - 5)^{5}). First, find (u') and (v'). For (u = 9x^{2}), using the power rule ((x^{n})'=nx^{n - 1}), we have (u'=9\times2x=18x). For (v=(x - 5)^{5}), using the chain rule ((f(g(x)))'=f'(g(x))\cdot g'(x)) where (f(u)=u^{5}) and (g(x)=x - 5). Then (f'(u) = 5u^{4}) and (g'(x)=1), so (v'=5(x - 5)^{4}\times1 = 5(x - 5)^{4}).
Step2: Substitute into the product rule formula
(y'=u'v+uv') (y'=18x\cdot(x - 5)^{5}+9x^{2}\cdot5(x - 5)^{4})
Step3: Factor out common terms
Factor out (9x(x - 5)^{4}) from each term. (y'=9x(x - 5)^{4}[2(x - 5)+5x]) Expand the expression inside the brackets: (2(x - 5)+5x=2x-10 + 5x=7x-10)
Answer:
(y'=9x(x - 5)^{4}(7x - 10))