Calculus I: Derivatives Exam Cram Sheet

This document serves as a high-intensity review for Differential Calculus, focusing on the mechanics, rules, and conceptual applications of derivatives.

## Topic Weighting

Based on standard Calculus I curricula, the distribution of derivative-related questions is typically:

## Key Concepts Summary

1. The Formal Definition

The derivative represents the instantaneous rate of change or the slope of the tangent line at a specific point.

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

2. Differentiability and Continuity

• Differentiability $\implies$ Continuity: If $f'(x)$ exists at $a$, $f$ is continuous at $a$.
• **Continuity $\not\implies Differentiability**: A function can be continuous but not differentiable (e.g., f(x) = |x|$ at $x=0$ due to a sharp corner).

3. Differentiation Strategy Flowchart

## Common Pitfalls

[!WARNING] Avoid these frequent exam errors:

• Constant Rule Confusion: Remember that $\frac{d}{dx}[C] = 0$, not $C$.
• The Forgotten Chain: Always check if the inner function has a derivative other than 1. $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$, not just $\cos(2x)$.
• Implicit Differentiation: Forgetting to attach $\frac{dy}{dx} when differentiating terms involving y$.
• Quotient Rule Signs: Mixing up the order of the numerator ($u'v - uv'$). The derivative of the top comes first!

## Mnemonics / Memory Triggers

• The Quotient Rule: "Low d-High minus High d-Low, square the bottom and away we go."
  • Formula: $\frac{v \cdot du - u \cdot dv}{v^2}$
• The Product Rule: "Left d-Right plus Right d-Left."
  • Formula: $f(x)g'(x) + g(x)f'(x)$
• Trig Sign Flip:
  • Functions starting with 'C' (Cosine, Cotangent, Cosecant) always have a negative derivative.

## Formula / Equation Sheet

General Rules

Trigonometric Derivatives

## Practice Set

1. Basic Power Rule: Differentiate $f(x) = 4x^5 - 3x^2 + 10$.
   • Answer: $f'(x) = 20x^4 - 6x$
2. Product Rule: Find $\frac{dy}{dx}$ for $y = x^2 \sin(x)$.
   • Answer: $y' = 2x\sin(x) + x^2\cos(x)$
3. The Chain Rule: Differentiate $y = (3x^2 + 1)^4$.
   • Answer: $y' = 4(3x^2 + 1)^3 \cdot (6x) = 24x(3x^2 + 1)^3$
4. Implicit Differentiation: Find $y'$ if $x^2 + y^2 = 25$.
   • Answer: $2x + 2yy' = 0 \implies y' = -\frac{x}{y}$
5. Tangent Line Equation: Find the equation of the line tangent to $f(x) = e^x$ at $x=0$.
   • Step 1: $f(0) = e^0 = 1$. Point is $(0,1)$.
   • Step 2: $f'(x) = e^x \implies f'(0) = 1$. Slope is 1.
   • Answer: $y - 1 = 1(x - 0) \implies y = x + 1$

[!TIP] Always simplify your exponents before differentiating! $f(x) = \sqrt{x} should be viewed as x^{1/2}$ to use the power rule easily.