Question

How do you differentiate `f(x)=5x^5 (5x+6)^4`?

Answer 1

Use the product rule and, because the second function involves an expression raised to a power, the power rule + chaon rule (Other called the general power rule)

I write the power rule in the form `(FS)' = F'S+FS'`, so I get:

`f'(x)=25x^4(5x+6)^4 + 5x^5 4(5x+6)^3*5`
And that's the calculus.

(It doesn't look like the book's answer? Well, we can do some algebra to simplify.)

`f'(x)= color(red)( [ color(black)(25x^4)color(blue)((5x+6)^4) + color(black)(100x^5 color(blue)((5x+6)^3))]`

`= 25 x^4 color(blue)((5x+6)^3) color(red)( [ color(black)color(blue)((5x+6)^1) + color(black)(4x^1)]`

`=25 x^4 (5x+6)^3 (9x+6)` And if you want to simplify further and rewrite, do so. (Although it's easy to find the zeros as written now.)

`f'(x)=75 x^4 (3x+2) (5x+6)^3`