Derivative Calculator
Calculate derivatives of functions step by step. Find first, second, and higher order derivatives with detailed explanations of differentiation rules applied.
Examples: x^3 - 2x^2 + x - 1, 5x^4 + 3x^2, 2x - 7
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About This Calculator
The derivative measures how a function changes as its input changes - it's the instantaneous rate of change or the slope of the tangent line at any point. This calculator helps you find derivatives of polynomial, trigonometric, exponential, and logarithmic functions with step-by-step explanations.
What is a Derivative? The derivative of a function f(x), denoted f'(x) or dy/dx, represents the rate at which y changes with respect to x. Geometrically, it's the slope of the function at any given point.
Formal Definition: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Why Derivatives Matter:
- Physics: velocity is the derivative of position, acceleration is the derivative of velocity
- Economics: marginal cost, marginal revenue, elasticity
- Engineering: rates of change, optimization problems
- Biology: population growth rates
- Machine Learning: gradient descent optimization
Common Derivative Notations:
- f'(x) - Lagrange notation
- dy/dx - Leibniz notation
- Df - Euler notation
- ẏ - Newton notation (dot notation, for time derivatives)
This calculator supports polynomials, trig functions, exponentials, and logs. For integration, see our Integral Calculator. For solving equations, try our Equation Solver.
How to Use the Derivative Calculator
- 1Select the function type (polynomial, trigonometric, exponential, or logarithmic).
- 2Enter the coefficients and parameters for your function.
- 3Choose the derivative order (1st, 2nd, 3rd, etc.).
- 4Optionally enter a point to evaluate the derivative at.
- 5Review the step-by-step solution showing rules applied.
- 6Check the final derivative expression.
- 7Use the evaluated value for specific calculations.
- 8Compare different orders to see the pattern.
- 9Apply the rules to similar problems.
- 10Verify results using differentiation rules manually.
Basic Derivative Rules
The fundamental rules for finding derivatives.
Constant Rule
d/dx[c] = 0
The derivative of any constant is zero.
- d/dx[5] = 0
- d/dx[π] = 0
Power Rule
d/dx[x^n] = n·x^(n-1)
The most frequently used rule.
- d/dx[x²] = 2x
- d/dx[x³] = 3x²
- d/dx[x^5] = 5x^4
- d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2) = 1/(2√x)
Constant Multiple Rule
d/dx[c·f(x)] = c·f'(x)
Constants factor out of derivatives.
- d/dx[3x²] = 3·2x = 6x
- d/dx[5x^4] = 5·4x³ = 20x³
Sum/Difference Rule
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
Differentiate term by term.
- d/dx[x² + 3x] = 2x + 3
- d/dx[x³ - x² + x] = 3x² - 2x + 1
Chain Rule
The chain rule handles composite functions (functions within functions).
The Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Or in Leibniz notation: dy/dx = (dy/du) · (du/dx)
How to Apply It
- Identify the outer function f and inner function g
- Differentiate the outer function, keeping inner unchanged
- Multiply by the derivative of the inner function
Examples
Example 1: (2x + 3)^5
- Outer: u^5, Inner: 2x + 3
- d/dx[(2x + 3)^5] = 5(2x + 3)^4 · 2 = 10(2x + 3)^4
Example 2: sin(3x)
- Outer: sin(u), Inner: 3x
- d/dx[sin(3x)] = cos(3x) · 3 = 3cos(3x)
Example 3: e^(x²)
- Outer: e^u, Inner: x²
- d/dx[e^(x²)] = e^(x²) · 2x = 2x·e^(x²)
Example 4: ln(x² + 1)
- Outer: ln(u), Inner: x² + 1
- d/dx[ln(x² + 1)] = 1/(x² + 1) · 2x = 2x/(x² + 1)
Product and Quotient Rules
Rules for products and quotients of functions.
Product Rule
d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Memory aid: "first times derivative of second plus second times derivative of first"
Examples:
- d/dx[x·sin(x)] = 1·sin(x) + x·cos(x) = sin(x) + x·cos(x)
- d/dx[x²·e^x] = 2x·e^x + x²·e^x = e^x(2x + x²)
- d/dx[(x+1)(x-1)] = 1·(x-1) + (x+1)·1 = 2x
Quotient Rule
d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²
Memory aid: "lo d-hi minus hi d-lo over lo squared"
Examples:
- d/dx[x/sin(x)] = [sin(x) - x·cos(x)] / sin²(x)
- d/dx[(x+1)/(x-1)] = [(x-1) - (x+1)] / (x-1)² = -2/(x-1)²
- d/dx[sin(x)/cos(x)] = [cos²(x) + sin²(x)] / cos²(x) = sec²(x)
When to Use Which
Use Product Rule when:
- Two functions multiplied together
- Neither is a simple constant
Use Quotient Rule when:
- Function is a fraction
- Can't simplify to remove division
Trigonometric Derivatives
Derivatives of the six basic trig functions.
Basic Trig Derivatives
| Function | Derivative |
|---|---|
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| cot(x) | -csc²(x) |
| sec(x) | sec(x)tan(x) |
| csc(x) | -csc(x)cot(x) |
Pattern Recognition
Sine and Cosine cycle:
- d/dx[sin(x)] = cos(x)
- d/dx[cos(x)] = -sin(x)
- d/dx[-sin(x)] = -cos(x)
- d/dx[-cos(x)] = sin(x)
- (returns to sin(x))
With Chain Rule
When trig functions have composite arguments:
- d/dx[sin(3x)] = 3cos(3x)
- d/dx[cos(x²)] = -2x·sin(x²)
- d/dx[tan(2x+1)] = 2sec²(2x+1)
- d/dx[sin²(x)] = 2sin(x)cos(x) = sin(2x)
Inverse Trig Derivatives
| Function | Derivative |
|---|---|
| arcsin(x) | 1/√(1-x²) |
| arccos(x) | -1/√(1-x²) |
| arctan(x) | 1/(1+x²) |
Exponential and Logarithmic Derivatives
Derivatives of exponential and log functions.
Natural Exponential
d/dx[e^x] = e^x
The only function that equals its own derivative!
With chain rule:
- d/dx[e^(2x)] = 2e^(2x)
- d/dx[e^(x²)] = 2x·e^(x²)
- d/dx[e^(sin(x))] = cos(x)·e^(sin(x))
General Exponential
d/dx[a^x] = a^x · ln(a)
Examples:
- d/dx[2^x] = 2^x · ln(2)
- d/dx[10^x] = 10^x · ln(10)
- d/dx[3^(2x)] = 3^(2x) · ln(3) · 2 = 2·ln(3)·3^(2x)
Natural Logarithm
d/dx[ln(x)] = 1/x
With chain rule:
- d/dx[ln(2x)] = 1/(2x) · 2 = 1/x
- d/dx[ln(x²)] = 2x/x² = 2/x
- d/dx[ln(sin(x))] = cos(x)/sin(x) = cot(x)
General Logarithm
d/dx[log_a(x)] = 1/(x·ln(a))
Higher Order Derivatives of e^x
All derivatives of e^x equal e^x:
- d/dx[e^x] = e^x
- d²/dx²[e^x] = e^x
- d^n/dx^n[e^x] = e^x
Higher Order Derivatives
Finding second, third, and higher derivatives.
Notation
- First derivative: f'(x), dy/dx, y'
- Second derivative: f''(x), d²y/dx², y''
- Third derivative: f'''(x), d³y/dx³, y'''
- nth derivative: f^(n)(x), d^n y/dx^n
Finding Higher Derivatives
Simply differentiate the previous derivative.
Example: f(x) = x^4
- f'(x) = 4x³
- f''(x) = 12x²
- f'''(x) = 24x
- f^(4)(x) = 24
- f^(5)(x) = 0
Physical Interpretations
| Derivative | Physical Meaning |
|---|---|
| Position s(t) | Location |
| s'(t) = v(t) | Velocity |
| s''(t) = a(t) | Acceleration |
| s'''(t) = j(t) | Jerk |
Patterns in Trig Functions
For sin(x):
- f(x) = sin(x)
- f'(x) = cos(x)
- f''(x) = -sin(x)
- f'''(x) = -cos(x)
- f^(4)(x) = sin(x) (cycle repeats!)
General formula:
- d^n/dx^n[sin(x)] = sin(x + nπ/2)
- d^n/dx^n[cos(x)] = cos(x + nπ/2)
Pro Tips
- 💡Always identify the function type before applying differentiation rules.
- 💡For composite functions, identify the "outer" and "inner" functions for chain rule.
- 💡Remember the power rule: bring down the exponent, then subtract 1 from it.
- 💡Check your answer by differentiating the derivative to get higher orders.
- 💡Use the sum rule to break complex functions into simpler terms.
- 💡Memorize the basic trig derivatives - they form the foundation for many problems.
- 💡The derivative of e^x is e^x - use this fact to simplify exponential problems.
- 💡Logarithmic differentiation can simplify products and quotients of powers.
- 💡Practice recognizing when to use product rule vs. chain rule.
- 💡Factor out constants before differentiating to simplify calculations.
- 💡Double-check signs, especially with trig and quotient rule derivatives.
- 💡Remember that d/dx[ln(x)] = 1/x works only for positive x.
Frequently Asked Questions
The derivative of x² is 2x. Using the power rule d/dx[x^n] = n·x^(n-1): d/dx[x²] = 2·x^(2-1) = 2x. This means the slope of the parabola y = x² at any point x equals 2x. At x = 1, the slope is 2; at x = 3, the slope is 6.
