how do you find f"(x) for f(x) = (4x+5) ^3

Question

anonymous · Answer

Is that all you were given?

campbell_st · Answer

this is the chain rule 

y = (f(x))^n      y' = n(f(x))^{n -1} * f'(x) 

simplify then repeat for y"

anonymous · Answer

yes, you need to find the first derivative and then the second derivative

campbell_st · Answer

thats correct

anonymous · Answer

yes, but that chain rule function doesnt help me, I have a hard time understanding it without the numbers

campbell_st · Answer

ok... 

so look at it this way 

1st derivative 

if you have y = x^n    y' = n *x^{n -1} 

does that make sense...?

campbell_st · Answer

so looking at your problem 

$$y = (4x + 5)^3.....y' = 3 \times (4x + 5)^2$$

does that make sense so far...

anonymous · Answer

yes

campbell_st · Answer

then you multiply by the derivative of 4x + 5 which is 4

so its 

$$y' = 3 \times(4x +5)^2 \times 4$$

giving you 

$$y' = 12(4x + 5)^2$$

does that make sense

anonymous · Answer

yes that does

campbell_st · Answer

then 2nd derivative is found in the same way, except you are multiplying by 12

so $$y" = 12 \times 2 \times(4x + 5) \times 4$$

just simplify that for the answer. 

with the can rule, whats inside the brackets never changes.  you just keep multiplying by its derivative

campbell_st · Answer

hope that helps

campbell_st · Answer

you may need to distribute for the final answer, but I wouldn't .

anonymous · Answer

thank you!