e^x is it's own derivative

[Euler]

I wanted to know if I demonstrated that e^x was it's own derivative appropriately. This is not a homework assignment, but I am curious to see if I correctly demonstrated this.

(Coincidentally, I wanted to check and make sure my latexing program worked.)

 

[Matt]

What is ln_e(e) supposed to mean?

 

[Euler]

It is the natural log (base e). log_b (b) = 1.

 

[stapel]

Please pardon my confusion, but, in going from "d(e^x)/dx" to "ln(e)", what happened to the variable?

Given y = e^x, might this be an instance in which logarithmic differentiation would be helpful? That is:

. . . . .y = e^x

. . . . .ln(y) = ln(e^x)

. . . . .ln(y) = xln(e)

. . . . .ln(y) = x

. . . . .(1/y)(dy/dx) = 1

. . . . .dy/dx = y

. . . . .dy/dx = e^x

I don't remember the sequence of proofs, though; I may be using something that needs to be proved before the above can be regarded as legitimate.

Hope that makes a little bit of sense.

Eliz.

 

[Euler]

Please pardon my confusion, but, in going from "d(e^x)/dx" to "ln(e)", what happened to the variable?

Given y = e^x, might this be an instance in which logarithmic differentiation would be helpful? That is:

. . . . .y = e^x

. . . . .ln(y) = ln(e^x)

. . . . .ln(y) = xln(e)

. . . . .ln(y) = x

. . . . .(1/y)(dy/dx) = 1

. . . . .dy/dx = y

. . . . .dy/dx = e^x

I don't remember the sequence of proofs, though; I may be using something that needs to be proved before the above can be regarded as legitimate.

Hope that makes a little bit of sense.

Eliz.

Oof I was wayy off. Could you perhaps delve into a bit more detail as to how this step came about from the previous?

. . . . .ln(y) = x

. . . . .(1/y)(dy/dx) = 1

Thanks, the rest of it made sense to me though!

 

[Gene]

She is using d(ln(u)) = 1/u*du on the left and
d(v) = dv on the right so dividing by dv
1/u*(du/dv)=1
multiplying by u gives
du/dv=u
d(ln(u)) should be in your table of derivitives. The only further expansion would be the proof of that.

 

[Euler]

Thanks for that Gene, I looked over my derivative table and found what you mentioned.