Calculus Derivatives Overview

Derivatives

Definition and Properties

• Definition: A derivative measures how a function changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero.
• Notation: Commonly denoted as ( f'(x) ) or ( \frac{dy}{dx} ).
• Properties:
  • If ( f(x) ) is continuous at ( x ), then ( f'(x) ) exists.
  • The derivative of a constant is zero: ( f'(c) = 0 ).
  • The derivative of ( x^n ) is ( nx^{n-1} ).
  • ( f'(x) ) indicates the slope of the tangent line to the curve at point ( (x, f(x)) ).

Higher-order Derivatives

• Definition: Higher-order derivatives are derivatives of derivatives.
• Notation:
  • Second derivative: ( f''(x) = \frac{d^2y}{dx^2} ).
  • Third derivative: ( f'''(x) = \frac{d^3y}{dx^3} ), and so forth.
• Interpretation:
  • The second derivative indicates concavity of the function.
  • Positive ( f''(x) ): Concave up (local minima).
  • Negative ( f''(x) ): Concave down (local maxima).

Rules of Differentiation

1. Sum Rule: ( (f + g)' = f' + g' )
2. Difference Rule: ( (f - g)' = f' - g' )
3. Product Rule: ( (fg)' = f'g + fg' )
4. Quotient Rule: ( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} )
5. Chain Rule: If ( y = f(g(x)) ), then ( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) )

Applications in Physics

• Velocity: The derivative of the position function with respect to time gives velocity.
• Acceleration: The derivative of the velocity function gives acceleration.
• Graphical Interpretation:
  • Increasing function: Positive derivative indicates motion in a positive direction.
  • Decreasing function: Negative derivative indicates motion in a negative direction.

Derivative Tests for Extrema

• First Derivative Test:

  • Identify critical points where ( f'(x) = 0 ) or ( f'(x) ) is undefined.
  • Determine the sign of ( f'(x) ) before and after the critical point.
    • If ( f' ) changes from positive to negative: local maximum.
    • If ( f' ) changes from negative to positive: local minimum.

• Second Derivative Test:

  • Compute ( f''(x) ) at the critical point ( x = c ).
    • If ( f''(c) > 0 ): local minimum.
    • If ( f''(c) < 0 ): local maximum.
    • If ( f''(c) = 0 ): test is inconclusive.

Definition and Properties

• A derivative quantifies the rate of change of a function as its input varies.
• Defined mathematically as the limit of the average rate of change as the interval approaches zero.
• Notated as ( f'(x) ) or ( \frac{dy}{dx} ).
• If a function ( f(x) ) is continuous at a point ( x ), then ( f'(x) ) exists.
• The derivative of a constant function is zero.
• For a power function, the derivative follows the formula: ( \frac{d}{dx}(x^n) = nx^{n-1} ).
• The first derivative ( f'(x) ) represents the slope of the tangent line to the function at any given point.

Higher-order Derivatives

• Higher-order derivatives refer to the derivatives of derivatives.
• Second derivative notated as ( f''(x) ) and represents ( \frac{d^2y}{dx^2} ).
• Third derivative notated as ( f'''(x) ) representing ( \frac{d^3y}{dx^3} ).
• The second derivative indicates the concavity of the function:
  • If ( f''(x) > 0 ), the function is concave up, suggesting a local minimum.
  • If ( f''(x) < 0 ), the function is concave down, suggesting a local maximum.

Rules of Differentiation

• Sum Rule: Derivative of a sum is the sum of the derivatives: ( (f + g)' = f' + g' ).
• Difference Rule: Derivative of a difference is the difference of the derivatives: ( (f - g)' = f' - g' ).
• Product Rule: For the product of two functions: ( (fg)' = f'g + fg' ).
• Quotient Rule: For the quotient of two functions: ( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} ).
• Chain Rule: For composite functions: if ( y = f(g(x)) ), then ( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ).

Applications in Physics

• Velocity is determined by taking the derivative of the position function concerning time.
• Acceleration is the derivative of the velocity function.
• A positive derivative indicates an increasing function and motion in a positive direction.
• A negative derivative indicates a decreasing function and motion in a negative direction.

Derivative Tests for Extrema

• First Derivative Test:

  • Identify critical points where ( f'(x) = 0 ) or ( f'(x) ) is undefined.
  • Analyze the sign of ( f'(x) ) around critical points.
  • A change from positive to negative indicates a local maximum.
  • A change from negative to positive indicates a local minimum.

• Second Derivative Test:

  • Evaluate ( f''(x) ) at the critical point ( x = c ).
  • If ( f''(c) > 0 ), a local minimum is present.
  • If ( f''(c) < 0 ), a local maximum is present.
  • If ( f''(c) = 0), the result is inconclusive.