Question

If f(t) =t^4 -2t^2+3t, find f'(t) and f"(t)

Answer 1

do you know how do derivatives?

Answer 2

yeah I think so!

Answer 3

Just do the derivative one time for the first f'(t). Then do the derivative of the f'(t) to get the equation for f"(t)

Answer 4

what would I put in for the integral?

Answer 5

? Why are you doing the integral?

Answer 6

When you see this "f'(t)" it means take the derivative of your original f(t) equation.

Answer 7

okay so whats the derivative? I'm sorry I'm just a little confused

Answer 8

The derivative is a measure of how a function changes as its input changes. Or we can say a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity.

In your function with respect to "t", the "t" being our "other quantity"

The formula you need for this type of derivative is:

\[f'(t) = nt ^{n-1}\] where "n" is the represent the variable or number in the exponent of each term of your equation f(t)

Answer 9

I will do the first term for you:
\[t^4\]if we were to take the derivative of this our process would be as so, we first bring down the "4" as stated in our formula up there:\[4t^4\] we then subtract one from the exponent, which is our "4" so we do that here:\[4t^{4-1}\] we then get this as our final derivative answer:\[4t^3\] this is only for the first term. Please do the rest and show me your answer.

Answer 10

okay so -4t^1? for the -2t^2 part and then for 3t would it be 3t^2?

Answer 11

right for the 2nd term but you stated "3t" as your original third term not "3t^2"

Answer 12

yeah 3t is the original, so would it just stay 3t?

Answer 13

No. For your third term, the original is 3t like you said, another way to represent this is:\[3t^1\] we bring down the "1" and subtract "1" from the exponent as so:\[(1)(3)t ^{1-1}\]any variable or number having an exponent of "0" reverts to "1" like this: \[x^0\]is just "1"

Answer 14

knowing this, what would your third term be?

Answer 15

3?

Answer 16

Excellent. Now you have solved the first derivative of f(t) or we could say we now have the equation of f'(t), but now they want the second derivative or f"(t). Can you take the derivative of the equation you just got?

Answer 17

yes i think so! except for the lonely 3 what would you do for that?

Answer 18

When you take a derivative of a constant, or a lonely number as you stated it. It automatically reverts to 0 since there's no change with respect to some other quantity right? or there's no "t" there. Solve it now and post your answer

Answer 19

12t^2-4?

Answer 20

YUP! Good job. You have solved the first and second derivative of the function "t" or f(t).

Answer 21

thank you so much! I really appreciate it!

Answer 22

You're welcome =)