06 Comparison
Table: Summary Table of Calculus Differentiation Rules and Formulations
1Constant Function
Mathematical Formula y = f(x)
- y = c
First Derivative dy/dx
- 0
Practical Exemplary Application
- d/dx [7] = 0
2Power Function
Mathematical Formula y = f(x)
- y = xn
First Derivative dy/dx
- n xn - 1
Practical Exemplary Application
- d/dx [x⁵] = 5 x⁴
3Constant Multiple
Mathematical Formula y = f(x)
- y = c u(x)
First Derivative dy/dx
- c u'(x)
Practical Exemplary Application
- d/dx [4 x³] = 12 x²
4Product of Functions
Mathematical Formula y = f(x)
- y = u × v
First Derivative dy/dx
- u v' + v u'
Practical Exemplary Application
- d/dx [x² (3x + 1)] = x²(3) + (3x + 1)(2x)
5Quotient of Functions
Mathematical Formula y = f(x)
- y = u / v
First Derivative dy/dx
- (v u' - u v') / v²
Practical Exemplary Application
- d/dx [(2x+1)/(x-3)] = [(x-3)(2) - (2x+1)(1)] / (x-3)²
6Extended Power Chain
Mathematical Formula y = f(x)
- y = [u(x)]n
First Derivative dy/dx
- n [u(x)]n-1 × u'(x)
Practical Exemplary Application
- d/dx [(x² + 5)⁴] = 4(x² + 5)³ × (2x) = 8x(x² + 5)³
7Square Root Chain
Mathematical Formula y = f(x)
- y = √u(x)
First Derivative dy/dx
- u'(x) / (2 √u(x))
Practical Exemplary Application
- d/dx [√(3x² + 1)] = 6x / (2 √(3x² + 1)) = 3x / √(3x² + 1)