SOLUTION: Find f ' (x) if f(x)=(4x-5)^3 x (5x^2+8)^3

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Question 1160094: Find f ' (x) if f(x)=(4x-5)^3 x (5x^2+8)^3
Found 2 solutions by MathLover1, math_helper:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=%284x-5%29%5E3+%2A+%285x%5E2%2B8%29%5E3......expand



f '
+f '


Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!

Another way....
+f%28x%29+=+%284x-5%29%5E3%2A%285x%5E2%2B8%29%5E3+
Use the product & chain rules:
Product rule d(f(x)g(x))/dx = f(x)g'(x) + f'(x)g(x)
Example: f'( +%283x%2B2%29%282x%5E2%2B1%29+) = +%283x%2B2%29%284x%29+%2B+%282x%5E2%2B1%29%283%29+ and then simplify if necessary. Yes, that simple!
"First times the derivative of the 2nd, plus the 2nd times the derivative of the first."


Chain rule (f(g(x)))' = f'(g(x))*g'(x)
Example: f' ( +%283x%5E2%2B2%29%5E4+ ) = +4%283x%5E2%2B2%29%5E3+%2A+%286x%29+
"Derivative of the expression times the derivative of what's inside."

f'(x) =
Factor out +%284x-5%29%5E2%2A%285x%5E2%2B8%29%5E2+
= +%284x-5%29%5E2%2A%285x%5E2%2B8%29%5E2+%2A+%28180x%5E2-150x%2B96%29+
= +highlight+%286%2A%284x-5%29%5E2%2A%285x%5E2%2B8%29%5E2+%2A+%2830x%5E2-25x%2B16%29+%29+