# Quick Math Differentiation

## Quick Math Differentiation

1
##### 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.