Derivatives 12th Grade

Mind Map: Derivatives

Central Idea

• Derivatives in Calculus

Main Branches

1. Definition of Derivative

• Limit of the difference quotient

• Notation: f'(x), dy/dx

• Geometric interpretation: slope of the tangent line

2. Basic Rules of Differentiation

• Power Rule

  • d/dx [x^n] = n*x^(n-1)

• Constant Rule

  • d/dx [c] = 0

• Sum Rule

  • d/dx [f(x) + g(x)] = f'(x) + g'(x)

• Difference Rule

  • d/dx [f(x) - g(x)] = f'(x) - g'(x)

• Product Rule

  • d/dx [f(x) * g(x)] = f'(x)g(x) + f(x)g'(x)

• Quotient Rule

  • d/dx [f(x) / g(x)] = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2

3. Common Derivatives

• Trigonometric Functions

  • d/dx [sin(x)] = cos(x)

  • d/dx [cos(x)] = -sin(x)

• Exponential Functions

  • d/dx [e^x] = e^x

• Logarithmic Functions

  • d/dx [ln(x)] = 1/x

4. Applications of Derivatives

• Finding Slopes

  • Tangent lines

• Optimization Problems

  • Maxima and minima

• Motion Analysis

  • Velocity and acceleration

5. Higher-Order Derivatives

• Second derivative: f''(x)

• Interpretation: concavity and inflection points

6. Implicit Differentiation

• Differentiating equations not solved for y

• Chain rule application

7. Related Rates

• Problems involving rates of change

• Setting up equations based on relationships

8. Graphical Interpretation

• Analyzing graphs using derivatives