What is a Derivative?
A derivative measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line at any point on a curve.
Lagrange Notation
Leibniz Notation
Euler Notation
Why Calculate Derivatives?
Derivatives are fundamental in calculus and have numerous practical applications across multiple disciplines:
Physics
Understanding motion and change in physical systems.
- Velocity is the derivative of position
- Acceleration is the derivative of velocity
- Force calculations in dynamics
Economics
Analyzing costs, revenue, and optimization.
- Marginal cost and marginal revenue
- Profit maximization
- Elasticity of demand
Engineering
Design optimization and system analysis.
- Optimization problems
- Signal processing
- Control systems design
Machine Learning
Training algorithms and model optimization.
- Gradient descent algorithms
- Minimizing loss functions
- Neural network backpropagation
About This Calculator
This derivative calculator helps you find derivatives of mathematical functions instantly with comprehensive step-by-step solutions. It supports a wide range of mathematical functions and differentiation techniques:
- Polynomial functions (x², x³, x⁴, etc.)
- Trigonometric functions (sin, cos, tan, cot, sec, csc)
- Inverse trigonometric functions (arcsin, arccos, arctan)
- Exponential and logarithmic functions (e^x, ln, log)
- Composite functions using the chain rule
- Higher-order derivatives (2nd, 3rd, up to 10th)
- 1. What is a Derivative?
- 2. How to Use This Calculator
- 3. Features
- 4. Frequently Asked Questions
- 4.1. What functions can this calculator differentiate?
- 4.2. How do I enter my function?
- 4.3. What is the chain rule?
- 4.4. Can I calculate higher-order derivatives?
- 4.5. Why is my result different from what I expected?
- 4.6. What does "Unknown identifier" error mean?
- 4.7. How accurate are the results?
- 4.8. Is my data saved or sent to a server?
How to Use This Calculator
Enter Your Function
Type your mathematical function in the input field. Use x as the variable. The calculator supports standard mathematical notation with flexible input syntax.
Input Syntax
| Operation | Symbol | Example |
|---|---|---|
| Addition | + |
x + 5 |
| Subtraction | - |
x - 3 |
| Multiplication | * or implicit |
2*x or 2x |
| Division | / |
x / (x+1) |
| Power | ^ |
x^3 |
Supported Functions
Trigonometric
sin(x), cos(x), tan(x), cot(x), sec(x), csc(x)Inverse Trigonometric
asin(x), acos(x), atan(x)Exponential
e^x, exp(x), 2^xLogarithmic
ln(x), log(x)Other Functions
sqrt(x), abs(x)Constants
e (≈2.718), pi (≈3.14159)Select Derivative Order
Choose the order of derivative you want to calculate:
- 1st - First derivative f'(x) - rate of change
- 2nd - Second derivative f''(x) - concavity and acceleration
- 3rd - Third derivative f'''(x) - rate of change of acceleration
- nth - Enter a custom order (up to 10th derivative)
Calculate
Click the Calculate Derivative button or press Enter. The result will display:
- The derivative formula in beautiful mathematical notation
- Step-by-step solution showing each differentiation step
- Applied differentiation rules (power rule, chain rule, etc.)
- Simplified final result
Features
Differentiation Rules
This calculator applies all standard differentiation rules automatically with precision. Understanding these rules is essential for mastering calculus:
Power Rule
For f(x) = x^n, the derivative is f'(x) = n·x^(n-1)
This is the most fundamental differentiation rule, used for any function with a variable raised to a power.
Examples:
- x³ → 3x²
- x^5 → 5x^4
- √x = x^(1/2) → (1/2)x^(-1/2) = 1/(2√x)
- 1/x = x^(-1) → -x^(-2) = -1/x²
Product Rule
For f(x) = u·v, the derivative is f'(x) = u'·v + u·v'
Used when differentiating the product of two functions. Each function is differentiated while the other remains unchanged, then the results are added.
Examples:
- x·sin(x) → 1·sin(x) + x·cos(x) = sin(x) + x·cos(x)
- x²·e^x → 2x·e^x + x²·e^x = (2x + x²)·e^x
- x³·ln(x) → 3x²·ln(x) + x³·(1/x) = 3x²·ln(x) + x²
Quotient Rule
For f(x) = u/v, the derivative is f'(x) = (u'·v - u·v') / v²
Used when differentiating a fraction where both numerator and denominator are functions of x.
Examples:
- x/(x+1) → [1·(x+1) - x·1] / (x+1)² = 1/(x+1)²
- sin(x)/cos(x) → [cos(x)·cos(x) - sin(x)·(-sin(x))] / cos²(x) = 1/cos²(x) = sec²(x)
- x²/(x-1) → [2x·(x-1) - x²·1] / (x-1)² = (x² - 2x) / (x-1)²
Chain Rule
For f(x) = g(h(x)), the derivative is f'(x) = g'(h(x))·h'(x)
The chain rule is essential for composite functions - functions within functions. Differentiate the outer function, then multiply by the derivative of the inner function.
Examples:
- sin(2x) → cos(2x)·2 = 2·cos(2x)
- e^(x²) → e^(x²)·2x = 2x·e^(x²)
- ln(x²+1) → [1/(x²+1)]·2x = 2x/(x²+1)
- (x²+3)^5 → 5(x²+3)^4·2x = 10x(x²+3)^4
Supported Functions
Trigonometric Derivatives
| Function | Derivative | Notes |
|---|---|---|
sin(x) |
cos(x) |
Most common trig function |
cos(x) |
-sin(x) |
Note the negative sign |
tan(x) |
sec²(x) |
Equivalent to 1/cos²(x) |
cot(x) |
-csc²(x) |
Equivalent to -1/sin²(x) |
sec(x) |
sec(x)·tan(x) |
Product of secant and tangent |
csc(x) |
-csc(x)·cot(x) |
Product with negative sign |
Inverse Trigonometric Derivatives
| Function | Derivative | Domain Restriction |
|---|---|---|
arcsin(x) |
1/√(1-x²) |
-1 < x < 1 |
arccos(x) |
-1/√(1-x²) |
-1 < x < 1 |
arctan(x) |
1/(1+x²) |
All real numbers |
Exponential & Logarithmic Derivatives
| Function | Derivative | Special Property |
|---|---|---|
e^x |
e^x |
Derivative equals itself |
a^x |
a^x·ln(a) |
General exponential form |
ln(x) |
1/x |
Natural logarithm (x > 0) |
log₁₀(x) |
1/(x·ln(10)) |
Common logarithm |
Higher-Order Derivatives
Calculate derivatives of any order from 1st to 10th. Higher-order derivatives reveal important properties of functions:
First Derivative
Second Derivative
Third & Higher
Step-by-Step Solutions
Every calculation shows the complete differentiation process with detailed explanations:
- Identification of which differentiation rule to apply
- Intermediate steps showing the application of each rule
- Simplification process from complex to final form
- Clear mathematical notation at each stage
LaTeX Rendering
Results are displayed in beautiful mathematical notation using LaTeX, making formulas easy to read and understand. Complex expressions with fractions, exponents, and special symbols are rendered professionally.
Hard to Read
- d/dx[x^2*sin(x)]
- = 2x*sin(x)+x^2*cos(x)
- Difficult to parse visually
Professional Display
- Beautiful mathematical symbols
- Proper fraction formatting
- Clear, textbook-quality output
Frequently Asked Questions
What functions can this calculator differentiate?
This calculator can differentiate a comprehensive range of mathematical functions:
- Polynomials - Any power of x (x², x³, x⁴, etc.)
- Trigonometric - sin, cos, tan, cot, sec, csc
- Inverse trigonometric - arcsin, arccos, arctan
- Exponential - e^x, a^x for any base
- Logarithmic - ln(x), log(x)
- Roots and radicals - sqrt(x), cube roots
- Absolute value - abs(x)
- Composite functions - Using the chain rule automatically
How do I enter my function?
Use standard mathematical notation with x as the variable:
- Use
^for powers:x^3for x³ - Use
*for multiplication or write implicitly:2*xor2x - Use parentheses for grouping:
(x+1)^2 - Function names:
sin(x),ln(x),sqrt(x)
Examples:
x^3 + 2*x^2 - 5*x + 1sin(x)*cos(x)e^(x^2)ln(x^2 + 1)
What is the chain rule?
The chain rule is used when differentiating composite functions - functions within functions.
Formula: If f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x)
In words: Differentiate the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function.
More examples:
- e^(x²) → e^(x²) · 2x
- ln(x³) → (1/x³) · 3x² = 3/x
- (x²+1)^5 → 5(x²+1)^4 · 2x
Can I calculate higher-order derivatives?
Yes! You can calculate derivatives up to the 10th order.
Select from the dropdown menu:
- 1st - First derivative (rate of change)
- 2nd - Second derivative (concavity, acceleration)
- 3rd - Third derivative (jerk in physics)
- nth - Custom order up to 10
The step-by-step solution shows each derivative calculation in sequence, so you can see how the function transforms at each level.
Why is my result different from what I expected?
The calculator automatically simplifies results to their most reduced form. Your result may look different but is mathematically equivalent.
Common simplifications:
- x/x becomes 1
- 0·x becomes 0
- x + x becomes 2x
- x² · x³ becomes x⁵
- sin²(x) + cos²(x) becomes 1
What does "Unknown identifier" error mean?
This error appears when you use a variable or function name the calculator doesn't recognize.
- Using variables other than x (like y, z, t)
- Misspelling function names (sine instead of sin)
- Using unsupported functions
- Missing parentheses: sin x instead of sin(x)
Solutions:
- Always use x as your variable
- Check function spelling: sin, cos, tan, ln, log, sqrt, etc.
- Include parentheses for all functions: sin(x), not sin x
- Click the ? icon for syntax help
How accurate are the results?
The calculator performs symbolic differentiation, meaning it applies differentiation rules exactly as you would by hand.
What this means:
- No rounding errors or approximations
- Exact symbolic expressions
- Same results as manual calculation
- Suitable for academic and professional use
Is my data saved or sent to a server?
No. All calculations are performed entirely in your browser using JavaScript.
Benefits:
- Complete privacy - no data transmission
- Works offline after initial page load
- Fast calculations - no server delays
- No account or login required
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