🔬 Higher Order Derivative Calculator

What are Higher Order Derivatives?

A higher order derivative is simply the result of differentiating a function repeatedly. If the first derivative, $f'(x)$ or $\frac{dy}{dx}$, gives the rate of change, the subsequent derivatives give the rate of change of the rate of change.

• Second Derivative: The derivative of the first derivative, denoted as $f''(x)$ or $\frac{d^2 y}{dx^2}$. It is used to determine the concavity of a function (whether the graph opens up or down) and identify inflection points.
• Third Derivative: The derivative of the second derivative, $f'''(x)$ or $\frac{d^3 y}{dx^3}$. In physics, the third derivative of position is known as **jerk** (the rate of change of acceleration).
• Nth Derivative: The result of differentiating $n$ times, written as $f^{(n)}(x)$ or $\frac{d^n y}{dx^n}$. This is especially important for Taylor Series expansions.

How Our Higher Order Derivative Calculator Works

Our tool uses the robust **math.js** library to perform symbolic differentiation. Unlike numerical methods that approximate the result, our calculator provides the exact algebraic expression for the derivative. The process involves:

1. Parsing: The input function is converted into an algebraic expression tree.
2. Iterative Differentiation: The tool applies the `math.derivative()` function iteratively, $n$ times, where $n$ is the order you specify. Each result is simplified before the next differentiation step begins.
3. Simplification: The final expression is simplified algebraically to provide the most concise result.

This method ensures accuracy and makes it easy to handle complex functions involving chain rule and product rule multiple times.

Applications of Higher Order Derivatives

Higher order derivatives have broad applications across science and engineering:

• **Physics:** The second derivative of position with respect to time is **acceleration**. The third derivative is **jerk**.
• **Economics:** The second derivative of a cost function helps determine if marginal costs are increasing or decreasing.
• **Curve Sketching:** The **Second Derivative Test** uses $f''(x)$ to classify critical points as local maxima or minima.

Start calculating your higher order derivatives today and master the deeper concepts of calculus!