Free collection of advanced calculus calculators for computing derivatives, integrals, and Riemann sums.
Calculate instantaneous rates of change, find areas under curves, and approximate integrals with step-by-step solutions.
Perfect for calculus students, engineers, and physicists - works on any device with no download required.
Enter a simple function using ^ for exponents (x^2) and * for multiplication (3*x)
Enter the variable you want to differentiate with respect to
• Use x^2 for x squared
• Use 3*x for 3 times x
• Use 2x + 1 for linear functions
• The calculator uses basic power rule differentiation
Select the type of calculus calculation you need from the dropdown menu (Derivative, Integral, or Riemann Sum)
Enter your mathematical function in the input field using standard mathematical notation
Specify the variable for differentiation/integration (e.g., x, t, etc.)
For integrals and Riemann sums, set the lower and upper bounds of integration
Click the 'Calculate' button to get instant results with detailed step-by-step explanations
Review the solution, graphs, and download or share your results if needed
Find velocity and acceleration from position functions in kinematics
Calculate marginal cost and marginal revenue from cost and revenue functions
Calculate the total distance traveled by a vehicle with varying velocity
Model population growth rates using differential calculus
Calculate the rate of change in compound interest accumulation
Complete calculus assignments with confidence by verifying derivatives and integrals. Perfect for AP Calculus, college calculus courses, and self-study. Get step-by-step solutions to understand the underlying concepts.
Solve real-world physics problems involving motion, forces, and energy. Calculate rates of change, optimize systems, and analyze continuous processes. Essential for mechanical, electrical, and civil engineering applications.
Perform marginal analysis, optimization problems, and calculate elasticities. Determine maximum profit, minimize costs, and analyze economic growth models. Critical for business analytics and economic forecasting.
Analyze experimental data, model natural phenomena, and solve differential equations. Calculate rates of reaction in chemistry, population dynamics in biology, and signal processing in computer science.
Demonstrate calculus concepts visually with interactive graphs. Help students understand the relationship between functions and their derivatives, the meaning of integrals, and the concept of limits.
Calculus is the mathematical study of continuous change, divided into two main branches: differential calculus and integral calculus. The derivative represents the instantaneous rate of change of a function at any given point, essentially the slope of the tangent line to the curve. When you calculate f'(x), you're finding how fast f(x) is changing with respect to x. Common differentiation rules include the power rule (d/dx[x^n] = nx^(n-1)), the product rule (d/dx[fg] = f'g + fg'), and the chain rule for composite functions.
Integration is the reverse process of differentiation, finding the accumulation of quantities or the area under curves. The definite integral ∫[a,b] f(x)dx represents the signed area between the curve f(x) and the x-axis from x=a to x=b. The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F(x) is the antiderivative of f(x), then ∫[a,b] f(x)dx = F(b) - F(a). Indefinite integrals find the general antiderivative family, always including a constant of integration C.
Riemann sums provide the foundation for definite integrals by approximating the area under curves using rectangles. The left endpoint, right endpoint, and midpoint methods offer different approximation strategies. As the number of rectangles increases and their width decreases, the Riemann sum approaches the exact integral value. This concept demonstrates how continuous accumulation can be understood as the limit of discrete sums, bridging finite and infinite processes in mathematics.
Forgetting the chain rule when differentiating composite functions
Always identify the inner and outer functions in composite expressions. Apply the chain rule: derivative of outer function evaluated at inner function, multiplied by derivative of inner function.
Missing the constant of integration in indefinite integrals
Remember that the antiderivative is a family of functions differing by a constant. Always add +C to indefinite integrals to account for all possible solutions.
Incorrect bounds when applying the Fundamental Theorem of Calculus
When calculating definite integrals, carefully apply F(b) - F(a). Double-check that you're subtracting the lower bound evaluation from the upper bound, not the reverse.
Confusing notation for derivatives and integrals
Use consistent notation: dy/dx, f'(x), or D[f(x)] for derivatives; ∫f(x)dx for indefinite integrals, and ∫[a,b]f(x)dx for definite integrals with proper bounds.
Not considering domain restrictions when simplifying expressions
Be aware of domain limitations, especially with logarithms, square roots, and rational functions. Check that your solutions are valid within the original function's domain.
Our calculators support polynomials, trigonometric functions (sin, cos, tan), exponential functions (e^x), logarithmic functions (ln, log), and their combinations using basic arithmetic operations.
Our Riemann sum calculator provides accurate approximations that improve with more subdivisions. For functions without elementary antiderivatives, numerical methods provide results accurate to multiple decimal places.
Yes! These calculators are perfect for checking homework answers, understanding solution methods, and learning calculus concepts. The step-by-step solutions help you understand the process, not just get the final answer.
Yes, all calculations include detailed step-by-step explanations showing which rules were applied, intermediate results, and the reasoning behind each mathematical operation.
Left endpoint uses the left point of each subinterval, right endpoint uses the right point, and midpoint uses the center. Midpoint typically provides better approximations, and all converge to the same value as subdivisions increase.
Currently, our calculators focus on single-variable calculus. For partial derivatives and multiple integrals, you would need specialized multivariable calculus tools.
No, your calculations are not saved permanently. All computations happen in your browser, ensuring complete privacy. Your work is cleared when you refresh the page.
Yes, you can download graphs as images and export calculation results as PDF files for reports, presentations, or homework submissions.