Question

Find :
`\mathsf {(dy)/(dx) (3x^(2) - 9x + 5)^(9)}`

Require thorough explanation. Thanks in advance!

Answer 1

`(dy)/(dx)=9(6x-9)(3x^2-9x+5)^8`

Explanation:

Given equation:

`y=(3x^2-9x+5)^9`

To differentiate the given equation, use the chain rule:

Chain Rule

`\textsf{If }\: y=f(u) \:\textsf{ and } \: u=g(x) \textsf{ then}:`

`(dy)/(dx)=(dy)/(du) * (du)/(dx)`

`\textsf{Find } \:(dy)/(dx)\: \textsf{ if } \:y=(3x^2-9x+5)^9`

`\textsf{Let }\:y=u^9 \: \textsf{ where }\: u=3x^2-9x+5`

Differentiate the two parts separately:

`\implies (dy)/(du)=9u^8 \quad \textsf{and} \quad (du)/(dx)=6x-9`

Put everything back into the chain rule formula:

`\begin{aligned}(dy)/(dx) & =(dy)/(du) * (du)/(dx)\\\\\implies (dy)/(dx) & = (6x-9) * 9u^8\\\\& = (6x-9) * 9(3x^2-9x+5)^8\\\\& = 9(6x-9)(3x^2-9x+5)^8\end{aligned}`

Answer 2

`\sf{\qquad\qquad\huge\underline{{\sf Answer}}}`

Let's solve ~

`\qquad \sf  \dashrightarrow \:\sf \: (d)/(dx) (3 {x}^(2) - 9x + 5) {}^(9)`

we know :

`\qquad \sf  \dashrightarrow \:\sf \: (d)/(dx) ({x}^(n)) = nx {}^(n - 1)`

So, by using the folowing property with chain rule ~

`\qquad \sf  \dashrightarrow \:\mathsf { 9(3x^(2) - 9x + 5)^(8)} \sdot(6x - 9)`