Question

derivative of integral

Answer 1

\[\int\limits_{3}^{x^3}lntdt\]

Answer 2

the derivative of the integral is the integrand
but in this case the upper limit of integration is \(x^3\) so you need the chain rule

Answer 3

replace \(t\) by \(x^3\) and then multiply by the derivative of \(x^3\) namely \(3x^2\)

Answer 4

you get
\[\ln(x^3)\times 3x^2\] via the chain rule

Answer 5

Did anyone else get \[3x^2\ln (x^3)\] because thats what I got
but the answer sheet says 9x^2lnx...

Answer 6

oh ok im not the only one

Answer 7

like having
\[F(x)=\int\ln(t)dt\] where the derivative is \(\ln(x)\) and then asking what is the derivative of
\(F(x^3)\) which by the chain rule is \(F'(x^3)\times 3x^2\)

Answer 8

they used the fact that \[\ln(x^3)=3\ln(x)\]

Answer 9

just showing off the knowledge of the properties of the log

Answer 10

clear right?
\[3x^2\ln(x^3)=9x^2\ln(x)\]

Answer 11

ohhhh i get it know

Answer 12

thank you!

Answer 13

yw