## Quick Math Differentiation

##### Enter expression in this form

Example: 3x^2 + 8x -12

\[ f(x) = x^3 + 2x^2 + x \]

### Differential Calculus in Calculus

Differential calculus is a branch of calculus that focuses on the concept of a derivative. The derivative measures how a function changes as its input changes. It provides information about the rate of change or the slope of the function at any given point.

#### Definition

The derivative of a function \( f(x) \) with respect to \( x \) is defined as:

\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x} \]

This limit, if it exists, represents the instantaneous rate of change of \( f \) at \( x \).

#### Examples

Example 1: Derivative of a Polynomial Function

Function:

\[ f(x) = x^3 + 2x^2 + x \]

Derivative:

\[ f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(2x^2) + \frac{d}{dx}(x) \]

\[ f'(x) = 3x^2 + 4x + 1 \]

The derivative \( f'(x) \) gives the rate of change of the polynomial function \( f(x) \).

#### Example 2: Derivative of a Trigonometric Function

Function:

\[ f(x) = \sin(x) \]

Derivative:

\[ f'(x) = \frac{d}{dx}(\sin(x)) = \cos(x) \]

The derivative \( \cos(x) \) represents the rate of change of the sine function \( \sin(x) \).

#### Example 3: Derivative of an Exponential Function

Function:

\[ f(x) = e^x \]

Derivative:

\[ f'(x) = \frac{d}{dx}(e^x) = e^x \]

The derivative \( e^x \) shows that the rate of change of the exponential function \( e^x \) is the same as the function itself.

Applications

– Physics: Derivatives are used to calculate velocity and acceleration.

– Economics: Derivatives help in finding marginal cost and revenue.

– Engineering: Derivatives are used in optimizing systems and processes.

Differential calculus is a powerful tool for analyzing and understanding the behavior of functions and their rates of change.