## Definition of Differentiation

**Differentiation** means the rate of change between two individual quantities or values. The ratio of the small change in one value is dependent on the first value given in the function. The basic formula for differentiation is** dy/dx**, where **y=f(x)**.

## Differential vs. Derivative

Differential and Derivative are two intimately connected terms in calculus. The term derivative means the rate of change of one variable with respect to another one. Here, variables are the changing entities.

the differential equation defines the relationship between the variables and derivatives.

## Rules of Differentiation

When a differentiation point is **0**, the function remains continuous. Otherwise, for each interval of position, it sets a new product relating to the values. The differentiation rules are:

**1. Constant Rule**: **d[C]/dx=0**

**2. Power Rule**: **dx^n/dx=nx^n-1**

**3. Product Rule**: **d[f(x)g(x)]/dx=f’(x)g(x)+f(x)g’(x)**

**4. Quotient Rule**:** d/dx[f(x)/g(x)]=[g(x)f’(x)-f(x)g’(x)]/[g(x)]^2**

**5. Chain Rule**: **d/dx[f(g(x))]=f’(g(x))g’(x)**

### Step 1: Insert Horizontal Axis Values

- Insert the value of
**x**in**B5:B13**. - Make sure to enter the starting point
**0**. - Insert the value of
**n**.

### Step 2: Find Vertical Axis Values

To calculate the value of **y** for each value of **x, **use this function:

`y=x^n`

- Enter this formula in
**C5**.

`=B5^$E$5`

- Press
**Enter**. - You will see the first output of
**y**.

- Drag down the
**AutoFill**to apply the formula to the rest of the cells.

### Step 3: Calculate Differentiation

- Enter this formula in
**D5**.

`=(C6-C5)/(B6-B5)`

**dy** is the difference between the last value and the immediately previous value of the **column y**. (**dx **is a similar function)

- Press
**Enter**.

This is the output.

- Apply the same procedure to each set of values.

### Step 4: Prepare a Differentiation Graph

- Select
**B4:B13**and**D4:D13**.

- Go to the
**Insert**tab and click**Scatter**in**Charts**.

- Select
**Scatter with Smooth Lines and Markers**.

- Your initial graph based on the
**differential value vs. value of x**will be displayed.

This is the final output:

## Example: Calculate Velocity with Differentiation in Excel

- Enter the values of
**time**and**distance**in**columns B**and**C**.

- Enter this formula in
**D6**to calculate**delta t**.

`=B6-B5`

- Press
**Enter**. - Drag the bottom corner of
**D7**up to**D13**to find all values.

- Enter this formula in
**E6**.

`=C6-C5`

- Press
**Enter**. - Use the
**AutoFill**to drag down the formula.

- Enter this formula in
**F6**.

`=E6/D6`

- Use the
**AutoFill**to drag down the formula.

The values of velocity with differentiation are calculated.

- You can create a graph:

**Note**: As the

**time**and

**distance**are always

**0**in the beginning, therefore the initial value of velocity is

**0**as well.

## Things to Remember

- The constant value of differentiation is always
**0**in the**power rule**. - Make sure to insert a starting point.

**Download Workbook**

Practice here.

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