Differentiation Review for Exam

Differentiation Notes

The Derivative and the Tangent Line Problem

  • The derivative of a function measures how the function value changes as its input changes.

Fundamental Concepts

Derivative Functions

  • Derivative functions describe the rate of change of a function.

Basic Differentiation Rules

  • The primary tool for differentiation includes product and quotient rules, derivatives of trigonometric and exponential functions, and the chain rule.

Product and Quotient Rules

  • For functions f(x) and g(x):

    • Product Rule:
      d[f(x) imes g(x)] = f'(x) imes g(x) + f(x) imes g'(x)

    • Quotient Rule:
      rac{d}{dx}igg[ rac{f(x)}{g(x)}igg] = rac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Derivatives of Trigonometric and Exponential Functions

  • Key derivatives to memorize:

    • rac{d}{dx}[ ext{sin}(x)] = ext{cos}(x)

    • rac{d}{dx}[ ext{cos}(x)] = - ext{sin}(x)

    • rac{d}{dx}[ ext{tan}(x)] = ext{sec}^2(x)

    • Exponential function:
      rac{d}{dx}[e^x] = e^x

The Chain Rule

  • If h(x) = f(g(x)), then
    h'(x) = f'(g(x))g'(x)

Derivatives of Inverse Functions

  • If f is a one-to-one function and has an inverse, then
    rac{d}{dx}[f^{-1}(x)] = rac{1}{f'(f^{-1}(x))}

Higher Derivatives

  • Notation and meaning:

    • First derivative: f'(x)

    • Second derivative: f''(x)

    • Third derivative: f'''(x)

    • n-th derivative: f^{(n)}(x)

Implicit Differentiation

  • Used when y is not isolated. Follow these steps:

    1. Differentiate both sides implicitly with respect to x.

    2. Collect terms and isolate y' to solve for it.

Tangent Lines

Definition and Calculation

  • The tangent line to the curve y = f(x) at the point (a, f(a)) is defined as:
    y - f(a) = f'(a)(x - a)

  • The slope (m) is the derivative at that point:
    m = f'(a)

Example

  1. Given the function f(x) = x² + 2, task to find the slope at (1,3):

    • m = lim_{x o 1} rac{f(x) - f(1)}{x - 1}

    • Slope calculated as 2.

  2. Find equation of tangent line:

    • y - 3 = 2(x - 1)

    • Simplified equation: y = 2x + 1

Applications of Derivatives

Physical Interpretations

  • The derivative f'(a) can represent rates of change in various fields:

    • Physics: s(t) represents position, s'(t) is velocity.

    • Economics: C'(t) represents cost changes over time.

Differentiability

Definition of Differentiability

  • A function is differentiable at a point if the derivative exists at that point.

  • Additionally, a function is differentiable on an interval if it is differentiable at every point in that interval.

Conditions for Non-Differentiability

  • The function may not be differentiable at:

    1. Points with discontinuities.

    2. Points with sharp corners (corners, cusps).

    3. Points with vertical tangents.

Summary of Differentiation Rules

  • Constant: d[c] = 0

  • Power Rule: d[x^n] = nx^{n-1}

  • Constant Multiple: d[c imes f(x)] = c imes f'(x)

  • Sum: d[f(x) + g(x)] = f'(x) + g'(x)

  • Difference: d[f(x) - g(x)] = f'(x) - g'(x)

Examples and Exercises

  1. Differentiate f(x) = 4x^2 - x + 2:

    • f'(x) = 8x - 1

  2. Find f'(x) for h(x) = (3x + 1)(x^2):

    • Apply product rule:

    • h'(x) = (3)(x^2) + (3x + 1)(2x)

  3. Example with trigonometric functions, such as f(x) = sin(x) + e^x to derive: f'(x) = cos(x) + e^x.

  4. Implicit differentiation example with x^2 + y^2 = 25. Differentiate both sides to find y':

    • 2x + 2y rac{dy}{dx} = 0
      ightarrow y' = - rac{x}{y}.

  5. Logarithmic Differentiation Example: For y = x^x, take the natural log of both sides and differentiate.

Conclusion

  • Mastering differentiation involves understanding functions’ behavior in different contexts, using various rules and interpretations to analyze and solve problems in mathematics and applied fields.