🎓 Mathematics • Calculus
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Derivative Calculator
Differentiates algebraic polynomials and trigonometric functions with step-by-step calculus derivations.
Polynomial Derivative: f(x) = axⁿ + bxᵐ + c
Calculates f'(x), f''(x), tangent line slope, and power rule step-by-step.
Presets:
First Derivative f'(x)
d/dx = 12x^2 − 10x
Second Derivative f''(x)24x − 10
Tangent Slope at x = 2m = 28
Step-by-Step Power Rule Derivation
1. Power Rule: d/dx(c·xⁿ) = c·n·xⁿ⁻¹
2. Term 1: d/dx(4x^3) = (4 × 3)x^(3−1) = 12x^2
3. Term 2: d/dx(-5x^2) = (-5 × 2)x^(2−1) = -10x^1
4. Constant term derivative: d/dx(7) = 0
5. Tangent slope evaluated at x = 2: f'(2) = 28
What is a Derivative?
A derivative measures the instantaneous rate of change of a function with respect to its independent variable.
Geometrically, f'(x) represents the slope of the tangent line to the curve at any point x.
Formula & Step-by-Step Calculation
d/dx (xⁿ) = n·xⁿ⁻¹, d/dx [sin(x)] = cos(x)
Fundamental calculus differentiation rules.
Worked Step-by-Step Examples
Example 1
d/dx (4x³ - 5x² + 7)
Solution: 12x² - 10x
• (4×3)x² - (5×2)x + 0 = 12x² - 10x
Common Real-World & Academic Use Cases
- ✓ Physics velocity from position (v = dx/dt)
- ✓ Economic marginal revenue curves
- ✓ Machine learning gradient descent
How to Use the Derivative Calculator
1
Select Function Type
Choose Polynomial or Trigonometric.
2
Input Coefficients
Fill in numbers.
3
View Derivative
Review f'(x) and tangent slope at chosen x.
Frequently Asked Questions
Q: What is the Power Rule?
Bring down the exponent as a multiplier, and reduce the power by 1.