Question

Help please!

Use the Second Fundamental Theorem of Calculus if needed to calculate the derivative expressed.

d/dx integral of (sin(t))/t dt from 2 to x

d/dx integral of (e^x)/x dt from x to squareroot of x

Answer 1

do you know about Second Fundamental Theorem of Calculus
?

Answer 2

If the function is continuous then the antiderivative of F (or the integral) is the function itself?

Answer 3

Or wait sorry the derivative of the antiderivative is the original function?

Answer 4

thats correct :)

Answer 5

so first part will be just sin x/x

know how to do 2nd part ?

Answer 6

so since it is dealing with e^x...

e^x then...is there something else?

Answer 7

there is a chain rule involved because the lower limit is not a constant, its a function of x

Answer 8

ok i am not sure how to do it with the second example :(

Answer 9

the chain rule says the derivative of the outer function then with the inner function

Answer 10

so it would be... e^x/x*(1)? so the answer would be e^x/x?

Answer 11

there's a lower limit too.... derivative of sqrt x is ....?

Answer 12

oh! ok ummm...

Answer 13

sqrt of x is x^(1/2) which means the derivative is (1/2)x^(-1/2)

Answer 14

thats correct!
so, we have
\(\Large e^x/x + (1/[2\sqrt x])(e^\sqrt x/\sqrt x)\)
try to simplify the 2nd term ?

Answer 15

why is (1/2)x^(-1/2) multiplied by (e^(sqrt(x))/sqrt(x)?

Answer 16

we're using this :
\(\Large \dfrac{d}{dx}\int \limits_{f(x)}^{g(x)}h(x)dx = H(g(x))g'(x)-H(f(x))f'(x)\)

Answer 17

so,
plugging in sqrt x for x in
(e^x)/x

Answer 18

ok so we have e^x/x+(1/2)x^(-1/2)(e^(sqrt(x))/x)

Answer 19

and that is the whole answer?

Answer 20

since e^x/x is for the x value and then (1/2)x^(-1/2)(e^(sqrt(x))/x) is for the (sqrt(x)) value?

Answer 21

yes, thats all correct! :)

Answer 22

okay great thank you :)

Answer 23

welcome ^_^