Derivative of the Exponent x

[Jason76]

The derivative of \(\displaystyle e^{-x}\) is \(\displaystyle -e^{-x}\) Shouldn't it be \(\displaystyle -e^{-x-1}\) according to the power rule?

 

[tkhunny]

No. That's why there is an exponent rule and logarithmic differentiation. The power rule does NOT apply to the variable of interest in the exponent.

If all else fails, there is a limit definition of a derivative. You can ALWAYS settle the question with the definition. Wait, maybe we should try that BEFORE "all else fails"!

Can you do it?

 

[Jason76]

No. That's why there is an exponent rule and logarithmic differentiation. The power rule does NOT apply to the variable of interest in the exponent.

If all else fails, there is a limit definition of a derivative. You can ALWAYS settle the question with the definition. Wait, maybe we should try that BEFORE "all else fails"!

Can you do it?

Ok, got it. When the derivative is taken, the number to the left of the exponent is brought down to be next to the left of the base. However, unlike with the power rule, the exponent stays the same (no subtraction)

For instance

\(\displaystyle 2e^{2x}\) would become \(\displaystyle 2(2)e^{2x}\) which yields \(\displaystyle 4e^{2x}\)

or

\(\displaystyle e^{3x}\) yields \(\displaystyle 3e^{3x}\)

\(\displaystyle e^{cos x}\) yields \(\displaystyle -sinx e^{cos x}\) or commonly written as \(\displaystyle e^{cos x}-sinx\)

 

[MarkFL]

To differentiate \(\displaystyle f(x)=2^{2x}\), you may rewrite the function as:

\(\displaystyle f(x)=e^{\ln(2^{2x})}=e^{2x\ln(2)}\)

and so:

\(\displaystyle f'(x)=e^{2x\ln(2)}\cdot2\ln(2)=2\ln(2)2^{2x}\)

edit: nevermind, I see you have edited your post.:mrgreen:

 

[lookagain]

For instance

\(\displaystyle 2e^{2x}\) would become \(\displaystyle 2(2)e^{2x}\) which yields \(\displaystyle 4e^{2x}\)

or

\(\displaystyle e^{3x}\) yields \(\displaystyle 3e^{3x}\)

\(\displaystyle e^{cos x}\) yields \(\displaystyle > > > -sinx e^{cos x} < < < \) . . . This is the correct derivative, yes.

or commonly written as > > \(\displaystyle e^{cos x}-sinx\) ** < < . . . . False

edit:

\(\displaystyle [-sin(x)]e^{cos(x)} ** \ \ means \ \ -sin(x) \ \ multiplied \ \ by \ \ e^{cos(x)}.\)


But, ** above means \(\displaystyle sin(x)\) subtracted from \(\displaystyle e^{cos(x)}.\)


** Brackets were added.

 

[HallsofIvy]

\(\displaystyle e^{cos x}(-sin(x))\) would be correct.

 

[lookagain]

. . .
and so:

\(\displaystyle f'(x)=e^{2x\ln(2)}\cdot2\ln(2)=2\ln(2)2^{2x}\)

Also, \(\displaystyle [2\ln(2)]2^{2x} \ = \ [\ln(2)]e^{2x + 1}\)