Question

derivitive of e^(xlnx)

Answer 1

let u = xlnx

so d/dx of e^u = e^u(du/dx)

du/dx = x(1/x) + (lnx)

Answer 2

well e^x =1/x so would that be e^xlnx =1/x2?

Answer 3

e^lnx = 1/x sorry

Answer 4

what do you mean.

what is the derivative of e^x?

Answer 5

notice that the x and lnx are multiplying.

Answer 6

so your saying the derivative x(1/x) + ln(X)?

Answer 7

no.

Answer 8

The derivative of e^x = e^x

Answer 9

so lets say that x was complicate expression. so we call that expression u

then d/dx e^u = e^u (du/dx)

Answer 10

ok

Answer 11

so now when I said that let u=xlnx

then I took the derivative of u with respect to x

so du/dx = the derivative of the (xlnx) part

when we can plug that part back into the original statement,

Answer 12

so at the final answer you can just replace u with xlnx and du/dx with the derivative
x(1/x) +lnx

Answer 13

so your final answer is x(1/x)+lnx? still not fully following sorry

Answer 14

no remember that the e^u expression is still there

Answer 15

So the answer would be

[x(1/x)+lnx]e^xlnx

Answer 16

oh i see so its \[(x(1/x)+\ln(x))e^xlnx\]

Answer 17

correct?

Answer 18

not quite... the lnx expression is in the exponent. \[[x(1\frac{1}{x})+\ln(x)]e^{xlnx}\]

Answer 19

thats what i ment sorry didnt see it didnt go in exponent

Answer 20

you can actually simplify this to \[[1+\ln(x)]e^{xlnx}\]

Answer 21

i know if you have \[e^(2x) \it = 2e^(2x) \] so why for this problem do you not just have \[(\ln(x))e^(xln(x))\]

Answer 22

those 2x are supposed to be in exponents sorry

Answer 23

also the xlnx part at bottom

Answer 24

because remember that xlnx are derived using a different method

derivatve of g(x)f(x) = g(x)f'(x) +f(x)g'(x)

where x= g(x) and f(x)=lnx

Answer 25

so your final answer is (1+ln(x)) e^(xln(x))

Answer 26

yes

Answer 27

the only thing i dont know is where the 1 came from

Answer 28

well

remember we had

x(1/x)

we can write that as x/x right? which is 1.

Answer 29

where did the x(1/x) come from i cant seem to find it

Answer 30

look carefully at the derivative of xlnx

do you see the part

\[x(\frac{1}{x})\]

what happens when you multiply out x with 1/x ?

Answer 31

oh i see. i thought you had to do the product rule to get derivative of x(lnx)

Answer 32

yes

Answer 33

when i do the product rule i got ln(x)+x(1/x)