Calculate Derivatives | Step-by-Step | All Rules Explained
✓ 3 Orders - First, second, third derivatives
✓ All Rules - Power, sum, product, chain
✓ Step-by-Step - Complete solutions
✓ All Functions - Polynomial, trig, exp, log
Calculate Derivatives | Step-by-Step Solutions | Multiple Orders
Derivative: Measures rate of change of a function. Enter function using x as variable.
💡 Examples: x^3, 2x^2+3x+1, sin(x), cos(x), e^x, ln(x)
Supported: x^n (powers), +/- (addition/subtraction), sin(x), cos(x), tan(x), e^x, ln(x)
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Power Rule
d/dx(x^n) = n·x^(n-1)
Constant Rule
d/dx(c) = 0
Sum Rule
d/dx(f+g) = f' + g'
Trig Rules
d/dx(sin x) = cos x
d/dx(cos x) = -sin x
Calculate first, second, and third derivatives with complete step-by-step solutions showing each differentiation rule applied at every step.
Supports all major differentiation rules including power rule, sum rule, constant rule, and special function rules with clear explanations.
Handle polynomials, trigonometric functions, exponential functions, and logarithmic functions with automatic rule detection and application.
A Derivative Calculator is essential calculus tool computing derivatives mathematical functions measuring instantaneous rate change fundamental concept differential calculus derivatives representing how function changes responding input changes slope tangent line function graph velocity acceleration higher order rates change derivative operation denoted various notations prime notation f-prime x Leibniz notation dy over dx operator notation D subscript x applied functions finding derivatives process called differentiation fundamental operation calculus alongside integration derivatives appearing throughout mathematics physics engineering economics computer science any field involving rates change optimization motion analysis derivative calculation following differentiation rules power rule most basic d over dx x raised n equals n times x raised n minus one constant comes down power reduces one sum rule derivative sum equals sum derivatives allowing term-by-term differentiation polynomials constant rule derivative constant zero constants unchanging product rule quotient rule chain rule handling composite functions derivatives essential analyzing function behavior finding maximum minimum values determining increasing decreasing intervals calculating rates change solving optimization problems understanding motion velocity acceleration describing marginal concepts economics analyzing concavity inflection points making calculator indispensable tool students researchers professionals working calculus.
Our Free Derivative Calculator 2025 offers comprehensive advanced features exceptional functionality multiple derivative orders first derivative f-prime x representing instantaneous rate change slope tangent line velocity position second derivative f-double-prime x representing rate change first derivative concavity function acceleration velocity third derivative f-triple-prime x representing rate change acceleration jerk physics supporting polynomial functions standard form a-sub-n x-raised-n plus terms differentiated using power rule sum rule each term treated separately trigonometric functions sine cosine tangent standard derivatives sine becomes cosine cosine becomes negative sine tangent becomes secant-squared exponential functions e-raised-x unique property being own derivative logarithmic functions natural logarithm ln x derivative one-over-x step-by-step solutions showing complete working every differentiation displaying original function identifying function type applying appropriate differentiation rules explaining each step showing intermediate results final derivative result rule identification labeling each rule application power rule sum rule constant rule trigonometric rules helping users understand which rules apply why educational transparent calculation history tracking last ten derivative calculations reviewing previous work comparing different functions learning patterns random example generator providing sample functions testing calculator quick demonstrations learning different function types mobile responsive design adapting all screen sizes smartphones tablets desktops touch-friendly inputs readable displays ensuring accessibility anywhere anytime making most powerful comprehensive user-friendly derivative calculator available online completely free unlimited calculations supporting calculus education worldwide helping students learn master differentiation techniques.
| Rule Name | Formula | Example |
|---|---|---|
| Power Rule | d/dx(x^n) = nx^(n-1) | d/dx(x³) = 3x² |
| Constant Rule | d/dx(c) = 0 | d/dx(5) = 0 |
| Sum Rule | d/dx(f+g) = f' + g' | d/dx(x²+x) = 2x+1 |
| Sine | d/dx(sin x) = cos x | d/dx(sin x) = cos x |
| Cosine | d/dx(cos x) = -sin x | d/dx(cos x) = -sin x |
| Exponential | d/dx(e^x) = e^x | d/dx(e^x) = e^x |
| Logarithm | d/dx(ln x) = 1/x | d/dx(ln x) = 1/x |
Simple steps: 1) Enter function using x (examples: x^3, 2x^2+3x+1, sin(x), e^x). 2) Select derivative order (First, Second, Third). 3) Click Calculate. Shows: Complete step-by-step solution, each rule applied, original function, final result, detailed explanations. Supports: Powers, polynomials, trig, exponential, logarithmic. Example: x^3 → 3x^2 with Power Rule steps. Perfect for calculus homework and learning.
Power Rule: d/dx(x^n) = n·x^(n-1). Steps: 1) Identify power n. 2) Multiply by power (bring down). 3) Subtract 1 from power. Examples: d/dx(x^3)=3x^2, d/dx(x^5)=5x^4, d/dx(5x^3)=15x^2. Calculator shows: Original term, power identified, multiplication, new power, final result. Works for negative/fractional powers. Most used differentiation rule.
Polynomial uses Sum Rule + Power Rule: f(x)=ax^n+bx+c → f'(x)=nax^(n-1)+b. Steps: 1) Apply Sum Rule (term by term). 2) Use Power Rule on each term. 3) Constants become 0. Example: 2x^3+3x^2-5x+7 → 6x^2+6x-5. Calculator shows each term differentiation, rule application, final combined result. Works any degree polynomial.
Standard trig derivatives: d/dx(sin x)=cos x, d/dx(cos x)=-sin x, d/dx(tan x)=sec²x. Calculator shows: Original trig function, rule applied, transformation, result. Example: sin(x)→cos(x). For multiples: d/dx(3sin x)=3cos x. Second derivatives: d²/dx²(sin x)=-sin x. Used for: wave motion, oscillations, periodic phenomena. Calculator displays formula and step-by-step.
Higher-order derivatives differentiate repeatedly: Second f''(x): derivative of f'(x), represents concavity/acceleration. Third f'''(x): derivative of f''(x), represents jerk. Example: f(x)=x^4 → f'(x)=4x^3 → f''(x)=12x^2 → f'''(x)=24x. Trig: sin(x) → cos(x) → -sin(x) → -cos(x). Calculator shows each derivative calculation separately with rules and results.
Special rules: EXPONENTIAL: d/dx(e^x)=e^x (equals original), d/dx(ke^x)=ke^x. LOGARITHMIC: d/dx(ln x)=1/x, d/dx(log_a x)=1/(x·ln a). Examples: e^x→e^x, 3e^x→3e^x, ln(x)→1/x, 5ln(x)→5/x. Calculator shows function type, standard rule, formula, steps. Used in: growth/decay, compound interest, population, radioactive decay.
Sum Rule: d/dx[f+g]=f'+g'. Differentiate each term separately, then add/subtract. Works for multiple terms. Example: x^3+2x^2-5x+7. Term 1: d/dx(x^3)=3x^2. Term 2: d/dx(2x^2)=4x. Term 3: d/dx(-5x)=-5. Term 4: d/dx(7)=0. Combine: 3x^2+4x-5. Calculator breaks down each term, applies rules, combines results. Makes complex derivatives manageable.
Constant Rule: d/dx(c)=0. Constants don't change, slope is 0. Examples: d/dx(5)=0, d/dx(100)=0. Constant multiple: d/dx(c·f)=c·f' (constant stays). Examples: d/dx(5x^2)=10x, d/dx(3sin x)=3cos x. Complete: 3x^2+7 → 6x+0=6x. Calculator identifies constants, applies rules correctly, simplifies results.
Each order has different meaning: FIRST f'(x): Rate of change, slope, velocity. f'>0 increasing, f'<0 decreasing, f'=0 critical points. SECOND f''(x): Concavity, acceleration. f''>0 concave up, f''<0 concave down. THIRD f'''(x): Jerk, rate of change of acceleration. Example x^3: f'(x)=3x^2 (slope), f''(x)=6x (concavity), f'''(x)=6 (jerk). Calculator shows all three with complete steps.
100% free forever! All features: First, second, third derivatives, complete step-by-step solutions, all differentiation rules (power, sum, constant, trig, exp, log), rule identification, detailed explanations, calculation history (10 items), random examples, multiple function types, error handling, mobile responsive. No registration, payment, ads, downloads, limits. Works any device/browser. Perfect for students, teachers, calculus learning, homework, exams. Free for everyone worldwide!
Our Free Derivative Calculator 2025 is the most advanced comprehensive derivative calculation tool supporting first second third derivatives complete step-by-step solutions all differentiation rules power rule sum rule constant rule trigonometric rules exponential rules logarithmic rules polynomial differentiation trigonometric function differentiation exponential function differentiation logarithmic function differentiation calculation history rule identification detailed explanations mobile responsive design completely free unlimited use no registration advertisements serving students teachers educators anyone learning calculus needing reliable accurate derivative calculator updated 2025 standards modern calculus education mathematical excellence supporting worldwide calculus learning.