Type in plain English or standard math notation. Voice input supported. Press Enter or click Solve. 🔍 Analyzing your problem… 🎯 Verified Answer 📚 Step-by-Step Tutorial How helpful was this solution? Thank you for your feedback! Essential Derivative Rules — Reference Guide with Worked Examples Understanding which differentiation rule to apply is the key skill in calculus. Below is a concise reference of every major derivative rule , with its formula and a worked example you can solve instantly in CalcMentor above. The most fundamental rule. If f(x) = xⁿ, then f′(x) = n · xⁿ⁻¹ . Multiply by the exponent, then reduce the power by 1. Example: d/dx(x⁵) = 5x⁴ Example: d/dx(x³) = 3x² For composite functions f(g(x)), the chain rule states: d/dx[f(g(x))] = f′(g(x)) · g′(x) . Differentiate the outer function, then multiply by the derivative of the inner function. Example: d/dx[sin(x²)] = cos(x²) · 2x = 2x·cos(x²) When two functions are multiplied: d/dx[f·g] = f′·g + f·g′ . Differentiate the first, keep the second; then keep the first, differentiate the second. Add both results. Example: d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x) For a ratio of two functions: d/dx[f/g] = (f′·g − f·g′) / g² . A helpful memory aid: "Low d-High minus High d-Low, over Low squared go." Example: d/dx[(x²+1)/(x−1)] = (x²−2x−1)/(x−1)² Exponential & Logarithm Rules Key formulas: d/dx[eˣ] = eˣ (unchanged) · d/dx[ln(x)] = 1/x · d/dx[aˣ] = aˣ·ln(a) . These appear in physics, economics, and biology models. Example: d/dx[e^(x²)] = 2x·e^(x²) (chain rule applied) Trigonometric Derivatives Essential trig rules: d/dx[sin(x)] = cos(x) · d/dx[cos(x)] = −sin(x) · d/dx[tan(x)] = sec²(x) . These underpin all wave, oscillation, and circular motion problems in physics and engineering. Example: d/dx[sin(x)·cos(x)] = cos²(x) − sin²(x) (product rule + trig) Use CalcMentor above to verify any formula with your own expressions. For in-depth written guides, visit our Learn Calculus hub, download free practice worksheets , or explore real-world applications of derivatives in physics, economics, and engineering. For focused practice, try our dedicated Chain Rule Calculator . How to Use CalcMentor: Complete Step-by-Step Guide Getting an accurate, verified answer is simple. Follow this guide to get the most reliable results every time — whether you are solving derivatives, integrals, limits, or algebraic equations. Type your math problem in the input box. You can use natural language or standard notation: Natural language: "Find the derivative of x squared plus 3x" Standard notation: "d/dx (x^2 + 3x)" Integrals: "integrate sin(x) from 0 to pi" or "∫sin(x)dx" Limits: "limit as x approaches 0 of sin(x)/x" Equations: "solve 2x^2 + 5x - 3 = 0" Simplify/Factor: "factor x^2 - 4" or "simplify (x^2-1)/(x-1)" Use ^ for exponents: x^2 , e^x , x^(n+1) Skip the * for multiplication: 2x , xsinx , 2xcosx — our parser handles implicit multiplication automatically Functions work without parentheses: sinx , ln2x , sincosx — all understood Use standard function names: sin , cos , tan , ln , sqrt , abs Press Enter or click SOLVE — Ctrl+Enter also works Click the 🎤 microphone to speak your problem aloud Click any ▶ Try button in the Derivative Rules section below to load and solve an example instantly Every problem goes through this intelligent pipeline to guarantee accuracy: Smart Input Understanding: Converts any format — natural language, Unicode symbols, even missing parentheses — into precise mathematical syntax using our advanced token-based parser Multi-Engine Computation: Multiple independent computation engines solve the problem separately. If two or more engines agree, the answer is marked as cross-verified with high confidence Priority-Based Verification: When engines disagree, the result from the highest-priority engine is used — never an AI guess. AI math solving is only used as an absolute last resort Step-by-Step Explanation: An AI tutor generates a numbered walkthrough with professionally typeset mathematical notation, explaining each transformation Verified Answer: The final answer displays in beautiful typeset mathematical notation. A green "Cross-verified" badge means multiple computation engines independently agreed on the answer. Interactive graph: If the expression can be plotted, a free graph appears automatically. Step-by-step tutorial: Click each step to expand it. Every step explains the reasoning with professionally rendered mathematical notation. Copy & Export: Easily 🧮 Copy as LaTeX, ⬇️ Download PDF, 🖨️ Print results directly, or 🎬 Export as Reel for video sharing! Generate Practice Tests: Automatically generate customized practice tests based on your exact problem to master the topic. CalcMentor is designed with redundancy. If one computation engine is temporarily slow, the system uses faster engines to deliver your answer without delay. Multiple engines cross-check every result — so you never get an unchecked or hallucinated answer. What Math Problems Can CalcMentor Solve? CalcMentor is a full-featured calculus solver and AI math solver that handles all core topics from high school through undergraduate level. Every solution is independently computed by multiple engines and verified before you see it — not a single AI guess. Power Rule: d/dx(x n ) = n·x n−1 Product Rule: d/dx(f·g) = f'·g + f·g' Quotient Rule: d/dx(f/g) = (f'·g − f·g')/g² Chain Rule: d/dx(f(g(x))) = f'(g(x))·g'(x) Implicit differentiation with multiple variables Second & higher-order derivatives ∫ Integrals & Limits Indefinite integrals with +C constant Definite integrals with bounds Integration by parts and substitution Limits: one-sided, at infinity, L'Hôpital's rule Trigonometric, exponential, logarithmic integrals

Rational function integration Integral Calculator → | Limit Calculator → 🔢 Algebra & Functions Solve: linear, quadratic, polynomial equations Simplify: rational expressions, radicals Factor: polynomials, quadratics, difference of squares Expand: polynomial and algebraic expressions Trig, inverse trig, hyperbolic, exponential, log functions Systems of equations with multiple variables Why Do You Need a Calculus Solver? Real-World Applications Derivatives and integrals are not just abstract math. They are the mathematical language of the real world. Students in physics, engineering, economics, biology, and computer science encounter calculus problems daily. CalcMentor is designed to help you solve these problems instantly so you can focus on understanding — not arithmetic. ⚡ Physics & Engineering Velocity is the derivative of position. Acceleration is the derivative of velocity. Calculating how a circuit's current changes over time, or how stress distributes in a beam — all require derivatives. Explore physics applications → 📈 Economics & Business Marginal cost, marginal revenue, and profit maximization all use derivatives. When a business asks "at what production level is profit maximum?", that is a calculus optimization problem solvable with differentiation . 🧬 Biology & Medicine Population growth models, drug concentration rates in the bloodstream, and the spread of diseases are all modeled with differential equations rooted in derivatives and integrals. Learn the foundations → 💻 Computer Science & AI Every neural network is trained using gradient descent — which is the process of computing derivatives (gradients) of a loss function. Without calculus, modern AI does not exist. Practice gradient problems → Frequently Asked Questions About CalcMentor Is CalcMentor really free? Yes, 100% free. No account required, no usage limits, no hidden fees, no credit card. CalcMentor is funded by non-intrusive display advertising so students everywhere can access it without barriers. How accurate are the results? CalcMentor uses multiple independent computation engines (not just one AI model) that solve every problem separately. When two or more engines independently reach the same answer, accuracy is effectively 100% for standard calculus and algebra. A confidence badge shows the verification status for every result. For highly complex or ambiguous inputs, always double-check against your course material. What derivative rules does CalcMentor support? CalcMentor supports all standard derivative rules: the power rule (d/dx xⁿ = n·xⁿ⁻¹), product rule (d/dx f·g = f'·g + f·g'), quotient rule (d/dx f/g = (f'·g − f·g')/g²), chain rule (d/dx f(g(x)) = f'(g(x))·g'(x)), trigonometric derivatives, exponential and logarithmic derivatives, and implicit differentiation. Does CalcMentor accept natural language input? Yes. You can type "derivative of x squared plus 3x", "differentiate sin(x^2)", or "find dy/dx of x^2 + y^2 = 25" and CalcMentor will understand you. Our advanced token-based parser handles implicit multiplication (2x, xsinx), missing function parentheses (sinx, ln2x), and Unicode symbols (², π, √). Can I use CalcMentor for homework? CalcMentor is designed as a learning tool. Use it to check your work, verify answers, and understand how solutions are derived. The step-by-step tutorials explain each transformation so you learn the underlying concepts — not just get the answer. Do I need to include the * for multiplication? No. CalcMentor's advanced parser automatically handles implicit multiplication. Type 2x instead of 2*x , xsinx instead of x*sin(x) , or 2xcosx instead of 2*x*cos(x) . Can I download my math solutions as a PDF or Video Reel? Absolutely. Unlike other AI math solvers that charge for premium features, CalcMentor allows you to instantly download your step-by-step solution as a high-quality PDF, copy the math as LaTeX code, or even export it as a video reel — 100% free. How does the Practice Test generator work? After solving any derivative, integral, or algebra problem, you can click "Generate Practice Test". CalcMentor's AI Math Tutor will instantly create 5 customized practice questions at varying difficulty levels based precisely on the topic you just studied. How is CalcMentor different from Mathway or Wolfram Alpha? The three main differences: (1) Free step-by-step solutions — Mathway hides step-by-step answers behind a $19.99/month paywall. CalcMentor shows full steps for free, always. (2) Multi-engine verification — Wolfram Alpha uses a single computation engine. CalcMentor uses three independent engines simultaneously; when they agree, the answer is cross-verified. (3) No account required — No sign-up, no email, no tracking. Open the page and solve. What is the derivative of sin(x), cos(x), and tan(x)? The three fundamental trigonometric derivatives are: d/dx[sin(x)] = cos(x) · d/dx[cos(x)] = −sin(x) · d/dx[tan(x)] = sec²(x) . These are foundational results that appear in physics, engineering, and signal processing. Type any of these into CalcMentor above to see the full step-by-step derivation. For more, visit our Trigonometric Derivative Calculator . Can CalcMentor solve implicit differentiation step by step? Yes. For implicit differentiation — equations like x² + y² = 25 where y cannot be isolated — type the full equation and include "dy/dx" in your query, such as "find dy/dx of x^2 + y^2 = 25". CalcMentor will differentiate both sides with respect to x, apply the chain rule to terms containing y, and solve for dy/dx step by step. Use our dedicated Implicit Differentiation Calculator for the most detailed solutions. What is the best free calculus solver online? CalcMentor is widely regarded as the best free calculus solver online in 2026 for three reasons: (1) it shows full step-by-step solutions for free — paid tools like Mathway charge $19.99/month for this, (2) it uses three independent computation engines to cross-verify every answer before you see it, and (3) it requires no account, no sign-up, and no email — open the page and solve immediately. Can CalcMentor solve any calculus problem step by step? Yes. CalcMentor is a full-featured AI calculus solver that handles all major calculus topics with complete step-by-step explanations: derivatives (power rule, chain rule, product rule, quotient rule, implicit differentiation, higher-order derivatives), definite and indefinite integrals (substitution, integration by parts, partial fractions), limits (one-sided, at infinity, L'Hôpital's rule). Type any calculus problem and get a verified, multi-step solution instantly. How does this calculus solver compare to Symbolab or Cymath? Compared to Symbolab and Cymath, CalcMentor offers three advantages: (1) No paywall — Symbolab limits free users and charges for full steps. CalcMentor is 100% free with no usage limits. (2) Multi-engine verification — CalcMentor cross-checks answers using three computation engines simultaneously. Symbolab and Cymath use a single engine. (3) No account required — Symbolab requires sign-up. CalcMentor works immediately, no registration needed. Does this AI math solver work for USA and UK calculus courses? Yes. CalcMentor's AI math solver is fully aligned with major curricula: USA (AP Calculus AB and BC, SAT Math, ACT Math, university calculus), UK (A-Level Mathematics and Further Mathematics, GCSE), Germany (Abitur Mathematik), Australia (VCE Mathematical Methods, HSC Mathematics Extension), and Canada (university-level calculus). The step-by-step format follows standard academic notation used in all these countries. What is a "derivative calculus solver" and how does it work? A derivative calculus solver is a tool that automatically finds the derivative of any mathematical function and shows each step of the process. CalcMentor's derivative calculus solver works by: (1) parsing your input into a precise mathematical expression, (2) applying the correct differentiation rules (power, chain, product, quotient, trig, exponential, logarithmic), (3) cross-verifying the result using three independent computation engines, and (4) generating a step-by-step tutorial explaining every transformation in plain English. Most math solvers either charge for step-by-step answers, require a sign-up, or use a single AI model that can hallucinate. CalcMentor is different: it uses multiple independent computation engines that solve every problem separately and cross-verify the result. You get the same answer quality as paid tools — completely free, forever. Every student gets full access to: 📋 Copy Result, 🧮 Copy as LaTeX, ⬇️ Download PDF, 🖨️ Print, 🎬 Export as Video Reel, and 📝 Generate Practice Tests — no account required. See how our verification works → CalcMentor vs Mathway vs Wolfram Alpha — Honest Comparison Choosing the right calculus solver matters. Here is a factual, side-by-side comparison of the three most popular tools students use for math help. CalcMentor was built specifically because paid tools like Mathway put step-by-step solutions behind a paywall that most students — especially those in developing countries — cannot afford. Our multi-engine verification approach means you also get a more reliable answer, not just a cheaper one. Every feature on this page is and will always remain free. Free Calculus Worksheets — Practice with Answer Keys → Download FREE derivative worksheets in PDF format. Includes chain rule worksheet, product rule worksheet, quotient rule worksheet with step-by-step solutions. Perfect for AP Calculus students and teachers. Organized by category: Core Derivatives, Limits, Exam Prep & AP Calculus. Welcome to CalcMentor! Get the best results with these simple tips: Your first computation may be slightly slower while the math engine loads. Math Detected From Image We extracted this math from your image. Please review and edit if needed: Edit the expression above if the extraction wasn't perfect, then click Solve.