Question

differentiate.
y = 9x²(x - 5)⁵
y =

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:

\(y'=9x(x - 5)^{4}(7x - 10)\)