Exponential Function derivatives
(Calculus AB)

Question

Exponential Function derivatives 
(Calculus AB)

darkknight · Answer

So the formulas...

For Base e:
$$d/dx (e^x)=e^x$$

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

darkknight · Answer

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

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