1. quick question...

well actually i have two quick questions. i am doing a problem and i wanted to make sure my thought process was correct.

1. when you have an integral, lets say inside the integral is 3*e^((1/2)*sin (2t)dt) with the limits running from 0 to x, if I wanted to take the derivative of this integral then that would simply remove the integral and leave me with the function at the endpoints of 0, and x? so i would have 3*e^((1/2)*sin (2(x)) dt - 3*e^((1/2)*sin (2(0)) dt ? is this correct?

2. my second question is on taking the derivative of the function y= e^(-(1/2)*sin(2x)). to do this derivative i could say -2*(1/2)*cos(2x)?

2. 1. You simply substitute x for t in the function (in other words just take the first half of your answer, and get rid of the dt).

2. Almost. There shouldn't be a minus sign, and the "e" disappeared.

3. ok, i forgot to include the e in the answer..sorry about that. now the minus sign i thought would carry through with the answer. the function is y = e^(-1)*(1/2)*sin(2x). I guess i am not understanding why the negative sign goes away.

4. The derivative of sin(2x) is 2cos(2x). No minus sign appears. The rest is just a constant and so stays where it is.

5. 1) That's the "fundamental theorem of Calculus": $\frac{d}{dx}\int_a^x f(t)dt= f(x)$.

2) The derivative of $e^{f(x)}$, with respect to x is $f'(x)e^{f(x)}$.

6. thanks halls, i got it. u guys are awesome!

7. Originally Posted by HallsofIvy

2) The derivative of $e^{f(x)}$, with respect to x is $f'(x)e^{f(x)}$.
As written, the question does not have $e^{f(x)}$ in it, just a constant. But perhaps the question is written incorrectly?

8. Originally Posted by DrSteve
As written, the question does not have $e^{f(x)}$ in it, just a constant. But perhaps the question is written incorrectly?
In the original post, the function was 3*e^((1/2)*sin (2t)dt) or $3e^{(1/2)sin (2t)dt}$ but in the repeat the outer parentheses were left off.

9. i am sorry let me retype the original problem again...

y= e^((-1/2)*sin(2x)) * (integral(3*e^((1/2)*sin(2t)) dt+ c1)) i hope this is better. the first part is y = e raised to the stuff, where the stuff is -1/2 times sin 2x, this is times the integral of

stuff, where the stuff equals 3 times e raised to the 1/2 times sin 2t, dt + c1. so to take the derivative of this function, i used the formula that halls posted earlier.

y' = would be just the product rule of the function. and after doing this problem i ordered some books on latex, i will work on a more suitable notation, that is more clearly understood.