This note has been used to help create the Power Rule wiki

Definition

Algebraic Functions

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 and Logarithmic Functions

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 \( f'(x) = \frac{1}{x} \).

Technique

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