🧮 Derivative Calculator – Symbolic, Numeric & Multivariable
The Derivative Calculator is a comprehensive calculus tool that computes derivatives of mathematical functions using the mathjs symbolic engine. Whether you need the symbolic expression of a derivative, the instantaneous slope at a specific point, higher-order derivatives, partial derivatives for multivariable functions, or implicit differentiation, this tool handles them all in one place.
📐 What Is a Derivative?
A derivative measures the instantaneous rate of changeof a function with respect to a variable. Geometrically, it represents the slope of the tangent line to the function’s graph at any given point. Formally, the derivative of f(x) is defined as:
f′(x) = lim(h → 0) [ f(x + h) − f(x) ] / hIn practice, derivatives are computed using symbolic rules rather than limits — the power rule, product rule, quotient rule, chain rule, and standard rules for trigonometric, exponential, and logarithmic functions.
⚙️ Five Calculation Modes
1. Symbolic Derivative
Enter any expression in terms of a single variable and receive the full symbolic derivative. For example, x^3 - 4*x + 7 produces f′(x) = 3x² − 4. The result is automatically simplified when the Simplify toggle is enabled.
2. Derivative at a Point
Evaluate the derivative numerically at a specific x value to obtain the instantaneous slope. When the evaluation is valid, the tool also produces the tangent line equation in the form y = m(x − x₀) + y₀ and overlays it on the graph.
3. Higher-Order Derivative
Set the order from 1 to 10 to compute repeated derivatives. The second derivative f″(x) describes acceleration and concavity; the third derivative relates to jerk in physics. The step-by-step panel shows each intermediate result.
4. Partial Derivative
For multivariable expressions like x^2*y + sin(x*y), select a variable and the tool differentiates with respect to it while treating all other variables as constants. This is the foundation of gradient computation and optimization in multiple dimensions.
5. Implicit Differentiation
When y cannot be isolated (e.g., x² + y² − 25 = 0), the tool computes dy/dx using the implicit differentiation formula:
dy/dx = −(∂F/∂x) / (∂F/∂y)where F(x, y) is the left-hand side of the equation set equal to zero.
📊 Interactive Graph
The dual-plot graph displays f(x) in blue and f′(x) in orange across your chosen x-range. In Point mode, the tangent line is added in green. You can adjust x min and x max to zoom into any region of interest — useful for spotting critical points, inflection points, and sign changes of the derivative.
+ - * / ^), parentheses, sin, cos, tan, sec, csc, cot, asin, acos, atan, exp, log (natural log), log10, sqrt, abs, and constants e and pi.📋 Key Differentiation Rules Applied
- Power Rule:
d/dx [xⁿ] = n·xⁿ⁻¹ - Product Rule:
d/dx [u·v] = u′v + uv′ - Quotient Rule:
d/dx [u/v] = (u′v − uv′) / v² - Chain Rule:
d/dx [f(g(x))] = f′(g(x))·g′(x) - Exponential:
d/dx [eˣ] = eˣ - Logarithm:
d/dx [ln x] = 1/x - Trig:
d/dx [sin x] = cos x,d/dx [cos x] = −sin x, etc.
💡 Practical Applications
Derivatives appear across science and engineering:
- Physics — velocity is the derivative of position; acceleration is the derivative of velocity.
- Economics — marginal cost and marginal revenue are derivatives of cost and revenue functions.
- Machine Learning — gradient descent relies on computing partial derivatives of loss functions.
- Engineering — control systems use derivatives to model dynamic responses.
⚠️ Limitations
This tool computes derivatives over the real numbers. Piecewise functions, absolute value at the origin, and expressions with branch cuts may produce unexpected results or domain warnings. Very complex nested expressions might not simplify fully — try expanding your expression manually if the result looks unwieldy.