Calculus Strategy Guide

1. The Power Rule (The Bread & Butter)

The most fundamental rule of differentiation. If you have a variable raised to a constant power, multiply the front by the power, and drop the power by 1.

\( \frac{d}{dx} x^n = n x^{n-1} \)

Example: \( \frac{d}{dx} 5x^3 = 15x^2 \)

2. The Chain Rule (The Onion Rule)

Used when dealing with composite functions (a function inside another function). Take the derivative of the "outside" function leaving the inside alone, then multiply by the derivative of the "inside" function.

\( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)

Example: For \( y = (3x+2)^5 \), the outside is \( u^5 \) and inside is \( 3x+2 \).
Result: \( y' = 5(3x+2)^4 \cdot 3 = 15(3x+2)^4 \)

3. The Product & Quotient Rules

When functions are multiplied together, you cannot just multiply their derivatives. You must use the product rule:

\( \frac{d}{dx} (fg) = f'g + fg' \)

For functions divided by each other, use the quotient rule (Remember: "Low D-High minus High D-Low, over Low squared we go!")

\( \frac{d}{dx} \left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2} \)

4. U-Substitution (The Reverse Chain Rule)

When integrating, if you notice a composite function where the derivative of the "inside" function is floating around in the equation, use U-Substitution.

Let \( u \) equal the inner function, take its derivative to find \( du \), and substitute them into the integral to simplify it into a basic Power Rule problem.

5. Integration By Parts

The integral equivalent of the Product Rule. Used when integrating the product of two unrelated functions (like an algebraic term and a trig term).

\( \int u \, dv = uv - \int v \, du \)

Pro-Tip (LIATE): Choose your '\( u \)' based on this hierarchy: Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential.