Question

Calculus Question!! MedaL!!

Answer 1

hint: fundamental theorem

Answer 2

\[\int\limits_{a}^{x} f(t)dt\]
\[f'(x) = f(x)\]

Answer 3

ok so all I have to do is evaluate f(x) and hat answer is equal to f'(x)?

Answer 4

yes right
but lets say if there is x^n \[n\cancel{= }1 \]
then you have to apply the chain rule \[\large\rm f'(x) = f(x^n) \cdot nx^{n-1}\]

Answer 5

ok wait so your saying if n does not equal 1 than apply chain rule?

Answer 6

no. we are still applying the chain rule
the derivative of x is just one
\[\int\limits_{2}^{x} \sin(2t) \rightarrow \sin(2x) \cdot 1\]
that's kinda confusing
so that's why i said assume n isn't equal to 1

Answer 7

All it's really saying is plug in x (from your limit) for t and that's your derivative

Answer 8

I'd suggest you finish the problem at hand first, before discussing the need to use the Chain Rule. The chain rule is not req'd in the current problem.

Answer 9

\[\int\limits_{2}^{x^2} \sin(t) dt \]
substitute x^2 for t \[\sin(\color{Red}{x^2}) \cdot \color{Red}{2x}\]
and then take the derivative of x^2

Answer 10

In the current problem you need only replace that variable 't' with x. Done.

Please, focus on finishing the current problem first.

Answer 11

i.e. \[h(x) = \int\limits_{1}^{x} (1+t^2) dt\] \[h'(x) = (1+x^2)\] if you want to skip the intermediate steps

Answer 12

you are still applying the chain rule but since it is just x ( and the derivative of x is just 1) you don't need to worry about anything for this question
but i was saying if the next question is asking to integrate from a to x(^ to some power greater than or less than 1) then the chain rule is really important

Answer 13

This info is helpful over the long term, but I think you (Nnesha) are confusing the issue by introducing this additional info before the original problem (not requiring chain rule) has been completed to the student's satisfaction. First things first, please.

Answer 14

I understand your advice about when to use the chain rule but after we have found that h'(x) = (1+x^2) what should be do next. Or is this the final answer?

Answer 15

For my problem, we're done we just have to replace t with x

Answer 16

Sorry to say sir, but i think chain rule isn't an additional info for this question
we are applying the chain rule here right ?

Answer 17

anyways, I apologize for the confusion.

Answer 18

Yeah, I think you're in the right track nnesha, you do apply it but the derivative of x is 1, if that's what you were thinking, so it doesn't really matter here :)

Answer 19

so technically you're applying chain rule for even this