Derivative of Exponential Functions

$$ 1. \ \frac {d}{dx} e^x = e^x $$

$$ 2. \ \frac {d}{dx} e^{f(x)} = e^{f(x)} \frac {d}{dx} f(x)$$

$$ 3. \ \frac {d}{dx} a^x = a^x \ln a $$

$$ 4. \ \frac {d}{dx} a^{f(x)} = a^{f(x)} \ln a \frac {d}{dx} f(x) $$

$$ 5. \ \frac {d}{dx} x^x = x^x (1+ \ln x) $$

Example:

$$ Differentiate \ f(x)=e^x+x $$

Solution:

$$ \frac {d}{dx} (e^x+x) = \frac {d}{dx} e^x + \frac {d}{dx} (x)$$

$$ \frac {d}{dx} (e^x+x) = e^x+1 $$