# Derivative Formula: Rules, List & Examples

The **derivative formula**, f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim\_{h \to 0} \dfrac{f(x+h) - f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​, measures a function's instantaneous rate of change at a single point, applied through standard rules — power, sum, product, quotient, and chain — together with the derivatives of trig, exponential, logarithmic, and inverse trig functions, with the chain rule and the product rule causing most student errors.

## What the derivative formula is

The **derivative formula** measures how fast a function's output changes when its input changes by a tiny amount. It is the foundation of differential calculus. For a function f(x), the derivative is written as f′(x) or dfdx\dfrac{df}{dx}. The derivative is defined by the limit:

> **The Derivative Formula (First Principles):**  
> f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim\\_{h \to 0} \dfrac{f(x+h) - f(x)}{h}

This is the formal definition. Most working calculus uses shortcut rules derived from it.

## Variable Key

| Symbol         | Meaning                                                            |
|----------------|--------------------------------------------------------------------|
| f(x)          | The original function                                              |
| f′(x)         | The derivative of f(x) — its rate of change                     |
| dfdx          | Leibniz notation for the same derivative                           |
| h              | A tiny change in x, shrinking toward zero                         |
| lim⁡h→0       | "As h approaches zero" — the limiting value                      |

## When To Use The Derivative Formula

Use a derivative whenever you need the **rate of change at a single instant** rather than over a stretch. Position changing into velocity. Velocity changing into acceleration. A drug concentration falling in the bloodstream. The slope of a curve at one specific point. Any time a problem asks _"how fast,"_ _"how steep,"_ _"the rate at which,"_ or _"the marginal,"_ a derivative is the tool.

## The Full List Of Derivative Formulas

The list below is the working toolkit. Most calculus problems are solved by recognising which formula applies and substituting.

### Basic derivative formulas

| Function       | Derivative     |
|----------------|-----------------|
| c              | 0               |
| x              | 1               |
| x^n           | n⋅x^(n-1)       |
| \sqrt{x}      | 1/(2√x)        |
| 1/x            | -1/x^2         |

### Trigonometric derivative formulas

| Function       | Derivative     |
|----------------|-----------------|
| sin⁡x          | cos⁡x          |
| cos⁡x          | -sin⁡x         |
| tan⁡x          | sec²x          |
| cot⁡x          | -csc²x         |
| sec⁡x          | sec⁡x⋅tan⁡x    |
| csc⁡x          | -csc⁡x⋅cot⁡x   |

### Exponential and logarithmic derivative formulas

| Function       | Derivative     |
|----------------|-----------------|
| e^x           | e^x            |
| a^x           | a^x⋅ln⁡a       |
| ln⁡x          | 1/x            |
| logₐx         | 1/(x⋅ln⁡a)     |

### Inverse trigonometric derivative formulas

| Function       | Derivative     |
|----------------|-----------------|
| sin⁡⁻¹x       | 1/√(1-x²)      |
| cos⁡⁻¹x       | -1/√(1-x²)     |
| tan⁡⁻¹x       | 1/(1+x²)       |
| cot⁡⁻¹x       | -1/(1+x²)      |
| sec⁡⁻¹x       | 1/|x|√(x²-1)   |
| csc⁡⁻¹x       | -1/|x|√(x²-1)  |

## Differentiation Rules

Rules combine the basic formulas above when functions are added, multiplied, divided, or nested.

**1. Constant Multiple Rule.**  
ddx[c⋅f(x)] = c⋅f′(x)

**2. Sum and Difference Rule.**  
ddx[f(x)±g(x)] = f′(x)±g′(x)

**3. Product Rule.**  
ddx[f(x)⋅g(x)] = f′(x)⋅g(x)+f(x)⋅g′(x)

**4. Quotient Rule.**  
ddx[f(x)/g(x)] = (f′(x)⋅g(x)−f(x)⋅g′(x))/[g(x)]²

**5. Chain Rule.**  
ddx[f(g(x)] = f′(g(x))⋅g′(x)

## Intuitive Derivation - Where The Formula Comes From

The slope of the line connecting two points on a curve approaches the instantaneous rate of change, which is the derivative.

## Worked Examples of Derivative Formula

### Example 1: Derivative of f(x)=x⁴
Apply the **power rule**:  
f′(x)=4⋅x³

**Final answer: f′(x)=4x³.**

### Example 2: Derivative of f(x)=3x⁵−2x³+7x−9
Apply the **sum/difference rule** term by term.

**Final answer: f′(x)=15x⁴−6x²+7.**

### Example 3: Derivative of f(x)=x²⋅sin⁡x
The correct tool is the **product rule**:  
f′(x)=(2x)(sin⁡x)+(x²)(cos⁡x)

**Final answer: f′(x)=2xsin⁡x+x²cos⁡x.**

### Example 4: Derivative of f(x)=sin⁡(x²)
The correct application is using the **chain rule**: 
f′(x)=2xcos⁡(x²)

**Final answer: f′(x)=2xcos⁡(x²).**

## Common Mistakes to Avoid

1. Forgetting the chain rule on composite functions.
2. Misapplying the product rule.
3. Dropping the negative sign on ddx[cos⁡x].
4. Power rule confusion when the exponent is the variable.