An algebraic function is a function that can be written using a finite number of the basic operations of arithmetic (i.e., addition, multiplication, and exponentiation). In order to take the derivative of these functions, we will need the power rule.

**The Power Rule:** The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

**Exponential Functions:** An exponential function with base \( e \) is its own derivative. That is to say, if \( f(x) = e^x \), then \( f'(x) = e^x \) as well.

**Logarithmic Function:** Logarithmic functions with base \( e \) have derivatives of the following form: if \( f(x) = \log_e x \), then \( f'(x) = \frac{1}{x} \).

## What is the slope of \( e^x \) when \( x = \ln 5 \)?

To find the slope, we find the derivative of \( e^x \). But that is simply \( e^x \) by the above rule. So the slope when \( x=5 \) is \(e^{\ln 5} = 5 \). \( _\square \)

## If \( f(x) = x^3 - 3x^4 \), what is \( f'(-4) \)?

Since we can take the derivative of each term separately,

\[ \begin{align} f'(x) &= 3x^{3-1} - 3(4)x^{4-1} \\ &= 3x^2-12x^3. \end{align} \]

Evaluating, \( f'(-4) = 3(-4)^2-12(-4)^3=48+768=816 \). \( _\square \)

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