# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/derivative-function/properties-of-derivative-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/derivative-function/properties-of-derivative-function/en.mdx

Output docs content for large language models.

---

export const metadata = {
    title: "Properties of Derivative Function",
    description: "Master essential derivative properties: power, product, quotient, and chain rules. Learn shortcuts to differentiate functions efficiently with clear explanations.",
    authors: [{ name: "Nabil Akbarazzima Fatih" }],
    date: "06/07/2025",
    subject: "Derivative Functions",
};

## Shortcuts to Derivatives

Finding the derivative of a function directly from its limit definition is a fundamental method, but it can be very long and tedious, especially for complex functions. The good news is, there are many shortcuts! Mathematicians have developed a set of practical rules known as the **properties of derivatives**.

These properties act like special tools that make the process of differentiating functions much faster and more efficient, allowing us to focus on the core of the problem.

## The Most Basic Rules

Let's start with a few fundamental rules that we will use frequently.

### Constant and Power Functions

The first two rules are the foundation for many derivatives.

1.  **Constant Function**: If a function is just a constant number, for example <InlineMath math="f(x) = k" />, its graph will be a straight horizontal line. A flat line has no slope at all, so its derivative is always zero.
    
    <BlockMath math="f(x) = k \implies f'(x) = 0" />

2.  **Power Rule**: This is a very powerful rule for functions of the form <InlineMath math="f(x) = ax^n" />. The method is simple: multiply the exponent <InlineMath math="n" /> by the coefficient <InlineMath math="a" />, then subtract one from the exponent.
    
    <BlockMath math="f(x) = ax^n \implies f'(x) = n \cdot ax^{n-1}" />

### Operations on Functions

What if we combine several functions? Suppose we have two functions, <InlineMath math="u(x)" /> and <InlineMath math="v(x)" />.

1.  **Constant Multiple**: If a function is multiplied by a constant, its derivative is that constant multiplied by the function's derivative.
    
    <BlockMath math="f(x) = k \cdot u(x) \implies f'(x) = k \cdot u'(x)" />

2.  **Addition and Subtraction**: This rule is very intuitive. The derivative of two functions that are added or subtracted is the sum or difference of their individual derivatives.
    
    <BlockMath math="f(x) = u(x) \pm v(x) \implies f'(x) = u'(x) \pm v'(x)" />

## Rules for Complex Functions

For more complex operations like multiplication, division, and function composition, we need special rules.

### Product Rule

When differentiating the product of two functions, we can't simply multiply their individual derivatives. The correct rule is as follows:

<BlockMath math="f(x) = u(x)v(x) \implies f'(x) = u'(x)v(x) + u(x)v'(x)" />

### Quotient Rule

Just like multiplication, division also has a special formula. Make sure the denominator function, <InlineMath math="v(x)" />, is not equal to zero.

<BlockMath math="f(x) = \frac{u(x)}{v(x)} \implies f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}" />

### Chain Rule

This rule is used for composite functions, or a "function within a function," like <InlineMath math="f(x) = [u(x)]^n" />. Imagine it like peeling an onion; we differentiate from the outermost layer inward. Differentiate the outer function first, then multiply by the derivative of the function inside.

<BlockMath math="f(x) = [u(x)]^n \implies f'(x) = n[u(x)]^{n-1} \cdot u'(x)" />