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

Mar 30, 2015

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 $\left(F S\right) ' = F ' S + F S '$, so I get:

$f ' \left(x\right) = 25 {x}^{4} {\left(5 x + 6\right)}^{4} + 5 {x}^{5} 4 {\left(5 x + 6\right)}^{3} \cdot 5$
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)]

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

$f ' \left(x\right) = 75 {x}^{4} \left(3 x + 2\right) {\left(5 x + 6\right)}^{3}$