Question

Derivatives of the exponent a^(x)

Answer 1

I have \[f(x) = 2^{x}-2^{-x}\]

could I write it as \[f(x) = 2^{x}- (2^{x})^{-1}\]

Answer 2

yes

Answer 3

If I take the derivative would it be \[f'(x) = 2^{x}\ln2 - [(-1)(\ln2)(2^{x})]\]

Answer 4

2nd term ? i think it isn't correct..

Answer 5

easier thing to do is
2^-x = (2^-1) ^x = (1/2) ^x

Answer 6

Oh alright then it would ne \[f'(x) = 2^{x}\ln2 - (\frac{1}{2})^{x}\ln(\frac{1}{2})\]

Answer 7

yes, and that comes out to be
2^xln 2 + 2^-x ln 2
= ln 2 (2^x+2^-x)

Answer 8

wait a minute where is your ln 1/2

Answer 9

ln 1/2 = - ln 2

Answer 10

did you use that rule before about -ln2

Answer 11

yes

Answer 12

so \[f'(0) = 2\ln2\]

Answer 13

correct.