# 3.2 - Derivative Rules for Exponential and Logarithmic Functions

Understanding the derivatives of exponential and logarithmic functions is crucial in calculus. Below, we'll explore the derivatives of $e^x$, $\ln(x)$, $a^x$, and $\log_a(x)$.

### 1. Derivative of $e^x$

The exponential function $e^x$ is unique in that its derivative is the same as the function itself.

$$\dfrac{d}{dx}e^x = e^x$$

### 2. Derivative of $\ln(x)$

The natural logarithm function, denoted as $\ln(x)$, is the inverse of $e^x$. Its derivative is represented as:

$$\dfrac{d}{dx}\ln(x) = \dfrac{1}{x}$$

### 3. Derivative of $a^x$ (where $a$ is a constant)

For any constant $a > 0$ and $a \neq 1$, the derivative of $a^x$ involves the natural logarithm of the base $a$.

$$\dfrac{d}{dx}a^x = a^x \ln(a)$$

### 4. Derivative of $\log_a(x)$

The derivative of the logarithm function with base $a$ (where $a > 0$ and $a \neq 1$) is given by:

$$\dfrac{d}{dx}\log_a(x) = \dfrac{1}{x \ln(a)}$$

### Summary

- The derivative of $e^x$ is itself.
- The derivative of $\ln(x)$ is $\dfrac{1}{x}$.
- The derivative of $a^x$ involves multiplying by $\ln(a)$.
- The derivative of $\log_a(x)$ is inversely proportional to $x \ln(a)$, expressed as $\dfrac{1}{x \ln(a)}$.

These rules facilitate solving calculus problems involving exponential and logarithmic functions with clarity and precision.

## Mnemonic Phrases for Remembering Derivative Formulas

To help remember the derivative rules for exponential and logarithmic functions, here are some memorable phrases:

### 1. Derivative of $e^x$

- **Phrase:** "E to the X, stays perplex, as its own derivative reflex."

### 2. Derivative of $\ln(x)$

- **Phrase:** "Log's nature's inverse, gives fraction's converse, one over X, no need to rehearse."

### 3. Derivative of $a^x$ (where $a$ is a constant)

- **Phrase:** "A to the X, multiplies by log base A, growing exponentially every day."

### 4. Derivative of $\log_a(x)$

- **Phrase:** "Log base A's descent, one over X times log makes it evident, inversely moving, never hesitant."

### Summary

- For **$e^x$**, remember it's unique, its own derivative, a fact quite sleek.
- **$\ln(x)$**'s derivative, simple and neat, **one over x**, a pattern to repeat.
- **$a^x$** grows with a twist, **multiplied by $\ln(a)$**, it can't be dismissed.
- And **$\log_a(x)$** changes its pace, **one over $x\ln(a)$**, in the derivative race.

These phrases are designed to be catchy and memorable, assisting in the recall of each function's derivative rule during exams or homework.