∂ Derivative Calculator: Complete Differentiation Guide
Master calculus differentiation with our advanced calculator featuring step-by-step solutions for all major differentiation rules.
Differentiation Rules Supported
Power Rule
d/dx(x^n) = n·x^(n-1)
Example: d/dx(x³) = 3x²Product Rule
(fg)' = f'g + fg'
For products of functionsQuotient Rule
(f/g)' = (f'g - fg')/g²
For ratios of functionsChain Rule
d/dx[f(g(x))] = f'[g(x)]·g'(x)
For composite functionsCommon Derivatives Reference
| Function | Derivative | Function | Derivative |
|---|---|---|---|
| Constant (c) | 0 | sin(x) | cos(x) |
| x^n | n·x^(n-1) | cos(x) | -sin(x) |
| e^x | e^x | tan(x) | sec²(x) |
| ln(x) | 1/x | sec(x) | sec(x)tan(x) |
| a^x | a^x·ln(a) | csc(x) | -csc(x)cot(x) |
Frequently Asked Questions
Q: What is a derivative?
A: A derivative measures the rate at which a function changes at any given point. Geometrically, it represents the slope of the tangent line to the function's curve. In physics, the derivative of position with respect to time is velocity. Derivatives are fundamental to calculus and have applications in science, engineering, economics, and many other fields.
Q: How do you calculate derivatives?
A: The method depends on the function type. Use: Power Rule for polynomials, Product Rule for multiplied functions, Quotient Rule for divided functions, Chain Rule for composite functions. For complex functions, apply multiple rules sequentially. Our calculator automatically identifies and applies the correct differentiation rules with step-by-step explanations.
Q: What are higher-order derivatives?
A: Higher-order derivatives are derivatives of derivatives. The second derivative f''(x) is the derivative of f'(x). Applications include: acceleration (second derivative of position), concavity analysis (second derivative tells if function curves up or down), inflection points (where concavity changes), and optimization problems in calculus.
Q: What is the chain rule?
A: The chain rule differentiates composite functions. If y = f(g(x)), then dy/dx = f'[g(x)] · g'(x). Example: For sin(x²), outer function is sin(u), inner is x². Derivative = cos(x²) · 2x = 2x·cos(x²). This is one of the most important and frequently used rules in calculus.
Q: Can this calculator handle implicit differentiation?
A: This calculator focuses on explicit functions where y is isolated. For implicit differentiation (equations like x² + y² = 1), you'd need specialized implicit differentiation tools. However, our calculator handles all standard explicit differentiation including trigonometric, exponential, logarithmic, and polynomial functions with complete step-by-step solutions.