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:
    $rac{d}{dx}\bigg[ rac{f(x)}{g(x)}\bigg] = rac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

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 $y'$:

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

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.