Calculus I Study Guide: Derivative Rules, Higher-Order Derivatives, Tangent Lines, Motion Analysis, and Product Rule

Tangent Lines and Parallel Slopes

• Parallel lines share the exact same slope ($m_1 = m_2$).
• Procedure to find the equation of a tangent line to $f(x)$ that is parallel to a given line $y = mx + b$:
  • Determine the required slope $m$ from the given parallel line.
  • Calculate the derivative $f'(x)$.
  • Set the derivative equal to the slope ($f'(x) = m$) and solve for $x_1$.
  • Find $y_1$ by evaluating the original function at $x_1$ ($y_1 = f(x_1)$).
  • Use point-slope form $y - y_1 = m(x - x_1)$ to write the line equation.
• Detailed Step-by-Step Example:
  • Function: $f(x) = x\times\frac{1}{2}$ rewritten as $f(x) = x\times\text{sqrt}(x) = x^{\frac{3}{2}}$
  • Derivative: $f'(x) = \frac{3}{2}x^{\frac{1}{2}} = \frac{3}{2}\sqrt{x}$
  • Given line to be parallel to: $y = 3x + 3$ (Slope $m = 3$)
  • Setting derivative equal to slope: $\frac{3}{2}\sqrt{x} = 3$
  • Solve for $x$:
  • Multiply both sides by $\frac{2}{3}$: $\sqrt{x} = 2$
  • Square both sides: $x_1 = 4$
  • Calculate $y_1$ using the original function: $f(4) = 4 \times \sqrt{4} = 4 \times 2 = 8$
  • Point coordinates: $(4, 8)$
  • Line equation calculation: $y - 8 = 3(x - 4)$$y - 8 = 3x - 12$$y = 3x - 4$

Differentiation Rules for Sums and Polynomials

• Sum Rule for Differentiable Functions:
  • If two functions $f(x)$ and $g(x)$ are differentiable, taking the derivative of their sum is equal to taking the derivative of each function individually: $\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$
• Derivatives of Constants:
  • The derivative of any constant function $c$ is zero: $\frac{d}{dx}[c] = 0$
  • Example: For $f(x) = x^2 - 7$, the derivative of $x^2$ is $2x$ and the derivative of $-7$ is $0$, yielding $f'(x) = 2x$.
• Multi-Term Polynomial Differentiation Example:
  • Function: $f(x) = x^7 + 13x^5 - 3x^4 + 12x^2 - 6x + 3$
  • Differentiating term-by-term using the Power Rule:
  • $\frac{d}{dx}[x^7] = 7x^6$
  • $\frac{d}{dx}[13x^5] = 65x^4$
  • $\frac{d}{dx}[-3x^4] = -12x^3$
  • $\frac{d}{dx}[12x^2] = 24x$
  • $\frac{d}{dx}[-6x] = -6$
  • $\frac{d}{dx}[3] = 0$
  • Final Combined Derivative: $f'(x) = 7x^6 + 65x^4 - 12x^3 + 24x - 6$

Algebraic Simplification Prior to Differentiation

• Algebraic manipulation must be performed before applying calculus derivative rules to avoid unnecessary complexity.
• Radical Expressions and Power Conversions:
  • Square roots must be converted into fractional exponents before applying the power rule: $\sqrt{x} = x^{\frac{1}{2}}$
  • Product rule for exponents when multiplying like bases: $x \times \sqrt{x} = x^1 \times x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}$
• Example 1 (Distributive Property before Differentiation):
  • Function: $y = \sqrt{x}(4 - x)$
  • Distribute algebraically before differentiating: $y = 4\sqrt{x} - x\sqrt{x} = 4x^{\frac{1}{2}} - x^{\frac{3}{2}}$
  • Differentiate using the power rule: $y' = 4 \times \left(\frac{1}{2}\right)x^{-\frac{1}{2}} - \frac{3}{2}x^{\frac{1}{2}} = 2x^{-\frac{1}{2}} - \frac{3}{2}x^{\frac{1}{2}}$
• Example 2 (Fraction Splitting vs. Exponent Distribution):
  • Function: $f(t) = \frac{2t^2 + 6t + 4}{\sqrt{t}}$
  • Method 1 (Splitting numerator over single denominator): $f(t) = \frac{2t^2}{t^{\frac{1}{2}}} + \frac{6t}{t^{\frac{1}{2}}} + \frac{4}{t^{\frac{1}{2}}}$
  • Subtract exponents ($t^{a-b}$): $f(t) = 2t^{2 - \frac{1}{2}} + 6t^{1 - \frac{1}{2}} + 4t^{-\frac{1}{2}} = 2t^{\frac{3}{2}} + 6t^{\frac{1}{2}} + 4t^{-\frac{1}{2}}$
  • Method 2 (Multiplying numerator by negative exponent): $\frac{1}{\sqrt{t}} = t^{-\frac{1}{2}}$$f(t) = t^{-\frac{1}{2}}(2t^2 + 6t + 4) = 2t^{\frac{3}{2}} + 6t^{\frac{1}{2}} + 4t^{-\frac{1}{2}}$
  • Differentiating the simplified expression: $f'(t) = 2\left(\frac{3}{2}\right)t^{\frac{1}{2}} + 6\left(\frac{1}{2}\right)t^{-\frac{1}{2}} + 4\left(-\frac{1}{2}\right)t^{-\frac{3}{2}}$$f'(t) = 3t^{\frac{1}{2}} + 3t^{-\frac{1}{2}} - 2t^{-\frac{3}{2}}$

Horizontal Tangent Lines

• A tangent line is horizontal if and only if its slope is zero ($m = 0$).
• Finding points on a curve $y = f(x)$ where tangent lines are horizontal requires:
  • Calculating the derivative $f'(x)$.
  • Setting $f'(x) = 0$ and solving for x$.\n - Substituting the resulting x$-values back into the original function$f(x)$to determine complete coordinates$(x, y).\n- Step-by-Step Example:\n - Function: y = x^4 - 4x^2 + 6\n - Derivative: y' = 4x^3 - 8x\n - Setting y' = 0:\n4x^3 - 8x = 0\n - Factor out common terms:\n4x(x^2 - 2) = 0\n - Roots for x:\n - 4x = 0 \implies x = 0\n - x^2 - 2 = 0 \implies x^2 = 2 \implies x = \pm\sqrt{2}\n - Calculate corresponding y$-coordinates by substituting into original function$y = x^4 - 4x^2 + 6:\n - For x = 0:\ny(0) = (0)^4 - 4(0)^2 + 6 = 6 \implies (0, 6)\n - For x = \sqrt{2}:\ny(\sqrt{2}) = (\sqrt{2})^4 - 4(\sqrt{2})^2 + 6 = 4 - 4(2) + 6 = 2 \implies (\sqrt{2}, 2)\n - For x = -\sqrt{2}:\ny(-\sqrt{2}) = (-\sqrt{2})^4 - 4(-\sqrt{2})^2 + 6 = 4 - 4(2) + 6 = 2 \implies (-\sqrt{2}, 2)\n - Points on the curve with horizontal tangent lines: (0, 6)$,$(-\sqrt{2}, 2)$, and$(\sqrt{2}, 2).\n\n# Derivatives of Trigonometric Functions and Higher-Order Derivatives\n\n- Basic Trigonometric Derivative Rules:\n\frac{d}{dx}[\sin(x)] = \cos(x)\n\frac{d}{dx}[\cos(x)] = -\sin(x)\n- Example Differentiation:\n - Function: y = 2\sin(\theta) + 6\cos(\theta)\n - First Derivative:\ny' = 2\cos(\theta) - 6\sin(\theta)\n - Second Derivative:\ny'' = -2\sin(\theta) - 6\cos(\theta)\n- Definition and Notation of Higher-Order Derivatives:\n - The second derivative f''(x)$or$\frac{d^2y}{dx^2} is the derivative of the first derivative.\n - Derivatives past the third derivative use numerical superscripts in parentheses to avoid excessive tick marks (e.g., f^{(4)}(x) for the fourth derivative).\n- Cyclic Behavior of Trigonometric Derivatives:\n - Derivatives of sine follow a repeating cycle of length 4:\n - f^{(0)}(\theta) = \sin(\theta)\n - f^{(1)}(\theta) = \cos(\theta)\n - f^{(2)}(\theta) = -\sin(\theta)\n - f^{(3)}(\theta) = -\cos(\theta)\n - f^{(4)}(\theta) = \sin(\theta)\n - Calculating the nth Derivative:\n - Divide n$by$4 and evaluate the remainder.\n - Example (Finding the 33rd derivative of \sin(\theta)):\n\frac{33}{4} = 8 \text{ remainder } 1\n - A remainder of 1 means f^{(33)}(\theta) = f^{(1)}(\theta) = \cos(\theta).\n- Polynomial Higher-Order Derivative Example:\n - First derivative: f'(x) = 4 - \frac{15}{4}x^{-\frac{1}{4}}\n - Calculating Second Derivative:\n - \frac{d}{dx}[4] = 0\n - \frac{d}{dx}\left[-\frac{15}{4}x^{-\frac{1}{4}}\right] = -\frac{15}{4} \times \left(-\frac{1}{4}\right)x^{-\frac{1}{4} - 1} = \frac{15}{16}x^{-\frac{5}{4}}\n - f''(x) = \frac{15}{16}x^{-\frac{5}{4}}\n\n# Physics Applications: Motion Functions\n\n- Position Function s(t)$: Represents the position of an object relative to an origin at time$t.\n- Velocity Function v(t): Represents speed with direction.\nv(t) = s'(t) = \frac{ds}{dt}\n - A negative velocity indicates motion in the downward or backward direction.\n- Acceleration Function a(t): Represents the rate of change of velocity.\na(t) = v'(t) = s''(t) = \frac{d^2s}{dt^2}\n - Positive acceleration represents speeding up; negative acceleration represents slowing down or accelerating downward.\n- Free-Fall Physics Example:\n - Position of an object dropped from a 100\,ft building:\ns(t) = -16t^2 + 100\n - Velocity function:\nv(t) = s'(t) = -32t\n - Acceleration function:\na(t) = v'(t) = -32\n- Multi-Step Motion Analysis Example:\n - Position function: s(t) = t^3 - 6t^2 + 9t\n - Step 1: Find Velocity and Acceleration functions:\nv(t) = s'(t) = 3t^2 - 12t + 9\na(t) = v'(t) = 6t - 12\n - Step 2: Evaluate acceleration after 2\,s$($t = 2):\na(2) = 6(2) - 12 = 0\,m/s^2\n - Interpretation: At t = 2\,s, the object is momentarily maintaining constant speed (not speeding up or slowing down).\n - Step 3: Find acceleration at the instant when velocity is zero for t > 2:\n - Set velocity to zero:\n3t^2 - 12t + 9 = 0\n - Divide by 3:\nt^2 - 4t + 3 = 0\n(t - 3)(t - 1) = 0\nt = 3 \quad \text{or} \quad t = 1\n - Select t = 3\,s$to satisfy condition$t > 2.\n - Calculate acceleration at t = 3:\na(3) = 6(3) - 12 = 18 - 12 = 6\n\n# The Product Rule\n\n- Purpose: Used to differentiate expressions where two functions of x are multiplied together.\n- Standard Formula:\n\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\n - Verbal statement: "Derivative of the first times the second, plus the first times the derivative of the second."\n- Algebraic Equivalence of Alternate Formula Formats:\n - Alternative ordering formulation:\n\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)\n - Because addition is commutative (a + b = b + a), both expressions are mathematically identical.\n- Step-by-Step Example:\n - Function: y = x\sin(x)\n - Identify components:\n - First function: f(x) = x \implies f'(x) = 1\n - Second function: g(x) = \sin(x) \implies g'(x) = \cos(x)\n - Apply Product Rule:\ny' = (1)\sin(x) + x\cos(x)\n - Simplified Final Answer:\ny' = \sin(x) + x\cos(x)$$