Calculate derivatives and integrals symbolically with step-by-step solutions and interactive graphs
Enter your function and calculation mode to get started. The calculator supports both symbolic and numeric differentiation and integration.
Derivatives and integrals are the two fundamental concepts of calculus, the mathematics of change and accumulation. Together, they form the backbone of modern science, engineering, economics, and data analysis, providing tools to model motion, optimize systems, compute areas and volumes, and analyze rates of change in any continuous process.
The derivative of a function f(x), denoted f'(x) or df/dx, measures the instantaneous rate of change of f with respect to x. Geometrically, it represents the slope of the tangent line to the curve at a given point. The derivative is defined as:
f'(x) = limh→0 [f(x+h) - f(x)] / h
Example: If f(x) = x², then f'(x) = 2x. This tells us that at x = 3, the slope of the tangent line is f'(3) = 6, meaning the function is increasing at a rate of 6 units of output per unit of input.
Applications: Derivatives are used to find velocities (rate of change of position), accelerations (rate of change of velocity), marginal costs and revenues in economics, optimization (finding maxima and minima), and analyzing curvature and concavity of functions.
The integral of a function f(x), denoted ∫f(x)dx, represents the accumulation of quantities or the area under the curve y = f(x). Integrals come in two flavors:
Example: If f(x) = 2x, then ∫2x dx = x² + C (indefinite), and ∫03 2x dx = [x²]03 = 9 - 0 = 9 (definite).
Applications: Integrals are used to compute distances from velocity, total accumulated quantities (mass, charge, revenue), areas and volumes, work done by forces, probabilities in continuous distributions, and solutions to differential equations.
The second derivative f''(x) measures the rate of change of the first derivative—i.e., the acceleration or concavity of the function. If f''(x) > 0, the function is concave up (curving upward); if f''(x) < 0, it is concave down. Third and higher derivatives measure even subtler aspects of change, such as jerk (rate of change of acceleration) in physics.
This theorem connects differentiation and integration, stating that integration and differentiation are inverse operations. Specifically, if F(x) = ∫ax f(t)dt, then F'(x) = f(x). This profound relationship allows us to evaluate definite integrals using antiderivatives and underpins much of applied mathematics.
This calculator performs both symbolic differentiation and integration, providing step-by-step solutions and interactive graphs. Follow these steps to compute derivatives and integrals:
x^3 + 2*x*sin(x) - e^x, ln(x) / (x^2 + 1).Output: The calculator displays a simplified symbolic expression (e.g., f'(x) = 3x² + 2cos(x) or ∫f(x)dx = x³/3 + C), an evaluated numeric result if a point or bounds were provided, and an interactive plot showing the original function f(x) overlaid with the derivative f'(x) or integral F(x). For definite integrals, the shaded region under the curve visually represents the computed area.
The calculator presents results in multiple formats to aid comprehension. Here's how to interpret each type of output:
| Output | Meaning & Interpretation |
|---|---|
| f'(x) | The first derivative, representing the instantaneous rate of change of f(x). Geometrically, f'(a) is the slope of the tangent line to y = f(x) at x = a. Positive f' indicates f is increasing; negative f' indicates decreasing; f' = 0 indicates a horizontal tangent (potential extremum). |
| f''(x) | The second derivative, representing the rate of change of f'(x), or equivalently, the acceleration or concavity of f(x). f'' > 0 means concave up (curving upward); f'' < 0 means concave down. Inflection points occur where f'' changes sign. |
| ∫f(x)dx | The indefinite integral, representing the general antiderivative of f(x)—a family of functions F(x) + C where F'(x) = f(x). The constant C accounts for all vertical shifts of the antiderivative. Use boundary conditions to determine a specific value of C if needed. |
| ∫abf(x)dx | The definite integral, evaluating to a specific number representing the net signed area between the curve y = f(x) and the x-axis from x = a to x = b. Positive f contributes positive area; negative f contributes negative area. Calculated via the Fundamental Theorem: F(b) - F(a). |
| Simplified Expression | The fully reduced symbolic result after applying differentiation or integration rules and algebraic simplification. This is the most compact and standard form of the answer, making it easier to interpret and use in further calculations. |
| Numeric Result | The evaluated value if you provided an x-value (for derivatives) or bounds [a, b] (for definite integrals). For derivatives, this is the slope at the given point. For integrals, this is the total area. Precision is controlled by the decimal places setting. |
| Interactive Graph | Visual representation of f(x) (blue) and f'(x) or F(x) (orange). For derivatives, observe where the derivative is zero (horizontal tangents on f) or changes sign (extrema). For integrals, the graph of F(x) shows accumulation; shaded regions (if definite) display the area being computed. |
Common Issues: If the result shows "undefined" or NaN, the function may not be defined at the evaluation point (e.g., ln(0), 1/0, or evaluating outside the domain). If the symbolic result is complex or unexpected, verify your input syntax—missing parentheses or incorrect operator precedence are common errors. For integrals, if no closed-form antiderivative exists (e.g., ∫e-x²dx), the calculator may return a numeric approximation or an expression involving special functions (erf, Si, etc.).
Common questions about derivatives, integrals, symbolic vs numeric computation, function support, and higher-order derivatives.
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