Derivative Calculator
Calculate Derivatives
Compute derivatives of functions with step-by-step solutions. Supports polynomials, trigonometric, exponential, and logarithmic functions.
Derivative Result
2x + 3
Rule Applied:
Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹
Sum Rule: d/dx[f(x) + g(x)] = f'(x) + g'(x)
Constant Rule: d/dx[c] = 0
Step-by-Step Calculation:
Function Analysis:
Critical Points: Solve f'(x) = 0 → x = -1.5
Increasing Interval: f'(x) > 0 when x > -1.5
Decreasing Interval: f'(x) < 0 when x < -1.5
Curvature: f''(x) = 2 > 0 → Concave up everywhere
Mathematical Notation: f'(x) = 2x + 3
LaTeX Format: \frac{d}{dx}(x^2 + 3x + 5) = 2x + 3
Application: Rate of change, optimization, tangent lines
The derivative represents the instantaneous rate of change of the function. For f(x) = x² + 3x + 5, the derivative f'(x) = 2x + 3 gives the slope of the tangent line at any point x.
What is a Derivative?
Derivative is a fundamental concept in calculus that 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. The derivative of a function f(x) with respect to x is denoted as f'(x) or df/dx.
Derivative Rules and Formulas
Power Rule
For polynomial terms
Example: d/dx[x³] = 3x²
Sum/Difference Rule
Derivative of sum
Linearity property
Product Rule
For multiplied functions
Leibniz rule
Quotient Rule
For divided functions
Division derivative
Derivative Rules by Function Type
1. Basic Rules
Fundamental derivative formulas:
• Identity: d/dx[x] = 1
• Power: d/dx[xⁿ] = n·xⁿ⁻¹
• Constant Multiple: d/dx[c·f(x)] = c·f'(x)
2. Trigonometric Functions
Derivatives of trigonometric functions:
• d/dx[cos(x)] = -sin(x)
• d/dx[tan(x)] = sec²(x)
• d/dx[cot(x)] = -csc²(x)
• d/dx[sec(x)] = sec(x)tan(x)
• d/dx[csc(x)] = -csc(x)cot(x)
3. Exponential & Logarithmic
Derivatives involving e and natural log:
• d/dx[aˣ] = aˣ·ln(a)
• d/dx[ln(x)] = 1/x
• d/dx[logₐ(x)] = 1/(x·ln(a))
Real-World Applications
Physics & Engineering
- Velocity & Acceleration: Derivative of position gives velocity, derivative of velocity gives acceleration
- Rate of change: How quickly physical quantities change over time
- Optimization problems: Finding maximum/minimum values in engineering design
- Electrical circuits: Current as derivative of charge, voltage relationships
Economics & Business
- Marginal analysis: Derivative of cost/revenue/profit functions
- Elasticity: Rate of change of demand with respect to price
- Optimization: Maximizing profit or minimizing cost
- Growth rates: Economic growth as derivative of GDP
Computer Science & Machine Learning
- Gradient descent: Derivatives optimize machine learning models
- Neural networks: Backpropagation uses chain rule of derivatives
- Computer graphics: Curve fitting and surface modeling
- Algorithm analysis: Rate of growth of functions
Biology & Medicine
- Population growth: Rate of change of population size
- Drug concentration: How quickly drugs are metabolized
- Enzyme kinetics: Reaction rates as derivatives
- Epidemiology: Rate of spread of diseases
Common Derivative Examples
| Function f(x) | Derivative f'(x) | Rule Used | Application |
|---|---|---|---|
| x² + 3x + 5 | 2x + 3 | Power, Sum | Quadratic motion |
| sin(x) | cos(x) | Trigonometric | Wave motion |
| eˣ | eˣ | Exponential | Continuous growth |
| ln(x) | 1/x | Logarithmic | Information theory |
Step-by-Step Derivative Calculation
Example 1: f(x) = 3x⁴ - 2x² + 7x - 5
- Apply power rule to each term: d/dx[3x⁴] = 12x³
- d/dx[-2x²] = -4x
- d/dx[7x] = 7
- d/dx[-5] = 0 (constant rule)
- Combine results: f'(x) = 12x³ - 4x + 7
- Simplify: f'(x) = 12x³ - 4x + 7
Example 2: f(x) = sin(x)cos(x)
- Identify: f(x) = g(x)·h(x) where g(x) = sin(x), h(x) = cos(x)
- Apply product rule: f'(x) = g'(x)·h(x) + g(x)·h'(x)
- g'(x) = cos(x), h'(x) = -sin(x)
- Substitute: f'(x) = cos(x)·cos(x) + sin(x)·(-sin(x))
- Simplify: f'(x) = cos²(x) - sin²(x)
- Alternative: f'(x) = cos(2x) using trigonometric identity
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Frequently Asked Questions (FAQs)
Q: What's the difference between derivative and differentiation?
A: Differentiation is the process of finding a derivative. The derivative is the result of differentiation. Differentiation is the action, derivative is the outcome.
Q: How do I find the second derivative?
A: The second derivative is the derivative of the first derivative. Calculate f'(x), then differentiate it again to get f''(x). It represents concavity and acceleration.
Q: What does the derivative represent graphically?
A: Graphically, the derivative at a point equals the slope of the tangent line to the function at that point. Positive derivative = increasing function, negative = decreasing.
Q: When should I use chain rule vs product rule?
A: Use chain rule for composite functions (function of a function): f(g(x)). Use product rule for multiplied functions: f(x)·g(x). Use quotient rule for divided functions: f(x)/g(x).
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