Derivative of a Function Calculator

Enter your function f(x) and the point x at which you want to calculate its derivative. Use standard JavaScript math syntax (e.g., Math.pow(x, 2) for x², Math.sin(x), Math.log(x) for natural log, Math.exp(x) for e^x).

Function f(x):

Point x:

Results:

Original Function Value f(x) at the point:

Numerical Derivative f'(x) at the point:

Understanding the Derivative of a Function

In calculus, the derivative of a function is a fundamental concept that measures the sensitivity of change of the function's value (output value) with respect to a change in its argument (input value). Essentially, it tells us how fast a function is changing at any given point.

What Does the Derivative Represent?

Rate of Change: The derivative represents the instantaneous rate of change of a function. For example, if a function describes the position of an object over time, its derivative would describe the object's instantaneous velocity.
Slope of the Tangent Line: Geometrically, the derivative of a function at a specific point is the slope of the tangent line to the function's graph at that point. A steeper tangent line indicates a larger derivative value, meaning the function is changing more rapidly.

Symbolic vs. Numerical Differentiation

There are two main ways to find a derivative:

Symbolic Differentiation: This involves applying a set of rules (power rule, product rule, chain rule, etc.) to find an exact algebraic expression for the derivative function. For example, if f(x) = x², its symbolic derivative is f'(x) = 2x. This method yields a new function that can then be evaluated at any point.
Numerical Differentiation: This method approximates the derivative at a specific point using the function's values at nearby points. It's particularly useful when the function is complex, known only through data points, or when symbolic differentiation is computationally too intensive.

How This Calculator Works (Numerical Differentiation)

This calculator uses a numerical approximation method to find the derivative. It employs the definition of the derivative as a limit:

f'(x) ≈ (f(x + h) - f(x)) / h

where h is a very small number (in this calculator, 0.000001). By calculating the function's value at x and at a point infinitesimally close to x (x + h), we can estimate the slope of the tangent line at x.

Examples of Using the Calculator

Let's look at some common functions and their derivatives:

Example 1: A Simple Polynomial

Function: f(x) = x² + 3x - 5
Point x: 2

Input in Calculator:
Function f(x): Math.pow(x, 2) + 3*x - 5
Point x: 2

Expected Symbolic Derivative: f'(x) = 2x + 3
Expected Value at x=2: f'(2) = 2(2) + 3 = 4 + 3 = 7

The calculator should output a value very close to 7.

Example 2: Trigonometric Function

Function: f(x) = sin(x)
Point x: Math.PI / 2 (approximately 1.570796)

Input in Calculator:
Function f(x): Math.sin(x)
Point x: 1.570796

Expected Symbolic Derivative: f'(x) = cos(x)
Expected Value at x=Math.PI/2: f'(Math.PI/2) = cos(Math.PI/2) = 0

The calculator should output a value very close to 0.

Example 3: Exponential Function

Function: f(x) = e^x
Point x: 1

Input in Calculator:
Function f(x): Math.exp(x)
Point x: 1

Expected Symbolic Derivative: f'(x) = e^x
Expected Value at x=1: f'(1) = e^1 = Math.E (approximately 2.718281)

The calculator should output a value very close to 2.718281.

Important Considerations

Accuracy: Numerical differentiation provides an approximation. The accuracy depends on the chosen value of h. A smaller h generally leads to better accuracy but can also introduce floating-point precision issues if too small.
Function Syntax: Ensure you use correct JavaScript syntax for mathematical operations (e.g., * for multiplication, Math.pow(base, exponent) for powers, Math.sin(), Math.cos(), Math.tan(), Math.log() for natural logarithm, Math.exp() for e^x).
Discontinuities: This method may produce misleading results for functions that are not differentiable at the given point (e.g., sharp corners, jumps).

This calculator is a helpful tool for quickly estimating the rate of change of a function at a specific point, providing a practical application of calculus concepts.