Question

Exponential Function derivatives
(Calculus AB)

Answer 1

So the formulas...

For Base e:
\[d/dx (e^x)=e^x\]

Any base:
\[d/dx(a^x)=a^x \times \ln(a)\]

Answer 2

Now lets get into some examples:

To find the first derivative of 2^x

\[d/dx(2^x) = 2^x \times \ln(2)\]


Now lets work with something harder

For example
\[f(x) = 3e^{-2x}\]
what is f' ???

Since this is base e, that's all we will have and we need chain rule (a tutorial on that might come too later)

so...

\[f'(x)=3e ^{-2x} \times -2\] where the -2 is the derivative of -2x, which is in the power
\[f'(x)=-6e ^{-2x} \]
Now lets find the second derivative of this

\[f''(x) = -6e ^{-2x}\times-2\]
\[f''(x) = 12e ^{-2x}\]