Question

How do you differentiate `f(x)= (x + 7)^10 (x^2 + 2)^7` using the product rule?

Answer 1

Product Rule:
If we have two functions `f(x)` and `g(x)`, then
`d/dx` `f(x)g(x)` = `f^'(x)g(x) + f(x)g^'(x)`

Chain Rule:
If we have two functions, `f(x)` and `g(x)`, then
`d/dx` `f(g(x))`=`g^'(x)•f^'(g(x))`

Original Equation:
`f(x)=(x+7)^10(x^2+2)^7`

Use the chain rule to find the derivative of the first equation:
`(x+7)^10`
`(1)•(10)(x+7)^9`
`10(x+7)^9`

Now multiply by the second function:
`10(x+7)^9 (x^2+2)^7`

Next, use the chain rule to find the derivative of the second equation:
`(x^2+2)^7`
`(2x)(7)(x^2+2)^6`
`(14x)(x^2+2)^6`

Multiply by the first function:
`(x+7)^10(14x)(x^2+2)^6`

Add together the two multiplied functions:
`(10(x+7)^9 (x^2+2)^7)`+`((x+7)^10(14x)(x^2+2)^6)`

Factor out `(x+7)^9` and `(x^2+2)^6`:
`{ (x+7)^9(x^2+2)^6} 10(x^2+2)+(x+7)(14x)`

Multiply and combine like terms for the factored section:
`10(x^2+2)+(x+7)(14x)`
`10x^2+20+14x^2+98x`
`24x^2+98x+20`

Put back the factors.
Your final answer should be:
`f^'(x)={ (x+7)^9(x^2+2)^6}(24x^2+98x+20)`