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.