Mathematics, Statistics & Geometry

Derivative Calculator

Calculate the symbolic derivative of any function instantly. Optionally evaluate the slope at a specific point for tangent line analysis.

Derivative f'(x)
6 * x + cos(x)
Calculation StepsFunction: f(x) = 3*x^2 + sin(x) Symbolic Derivative: f'(x) = 6 * x + cos(x)

Calculated locally in your browser. Fast, secure, and private.

Calculating the Rate of Change

The Derivative Calculator brings the power of a Computer Algebra System (CAS) directly to your browser. By performing exact symbolic differentiation, this tool provides the analytical formula for the derivative, as well as the numerical slope at any given point.

f′(x)=lim⁡h→0f(x+h)−f(x)h\begin{aligned} f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \end{aligned}

Where:
f'(x)=
The instantaneous rate of change of the function
f(x)=
The mathematical expression being differentiated
x=
The variable with respect to which we are differentiating

Common Derivative Rules Applied

Our engine automatically applies the fundamental rules of calculus to your input:

  1. Power Rule: $\frac{d}{dx} x^n = nx^{n-1}$
  2. Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
  3. Quotient Rule: For dividing functions.
  4. Chain Rule: For nested functions like $\sin(x^2)$.

Real-World Applications

  • Physics: If a function represents position, the derivative is velocity. The second derivative is acceleration.
  • Machine Learning: Derivatives (gradients) are used to minimize error functions in neural networks (Gradient Descent).
  • Economics: Calculating Marginal Cost or Marginal Revenue (the cost/revenue of producing one additional unit).
  • Engineering: Analyzing stress and heat transfer rates in materials.

Frequently Asked Questions

Geometrically, the derivative represents the slope of the tangent line to a curve at a specific point. Physically, it represents an instantaneous rate of change (like speed at an exact millisecond).

Symbolic differentiation applies calculus rules (like the power rule, chain rule, and product rule) to generate a new algebraic formula for the derivative, rather than just estimating a number.

The power rule states that the derivative of x^n is n*x^(n-1). For example, the derivative of x³ is 3x².

Use standard programming math notation. For example: 'x^2' for x-squared, 'sin(x)' for sine, 'sqrt(x)' for square root, and 'log(x)' for natural logarithm.

f'(x) gives you the general formula for the slope anywhere on the curve. Evaluating it at a point (e.g., x=2) gives you the exact numerical slope at that specific location.

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