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

Question

Elijah Abram · Accepted Answer

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#

Ezra Adams · Answer

undefined To differentiate the function $ f(x) = 5x^5 (5x+6)^4 $, you can use the product rule and the chain rule. The product rule states that if you have two functions multiplied together, the derivative of the product is the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule is used when differentiating compositions of functions.

So, applying the product rule and the chain rule, the derivative of $ f(x) $ is:

$$ f'(x) = 5 \cdot 5x^4 \cdot (5x + 6)^4 + 5x^5 \cdot 4(5x + 6)^3 \cdot 5 $$

$$ f'(x) = 125x^4(5x + 6)^4 + 100x^5(5x + 6)^3 $$

That's the derivative of the given function.