The Master Formula Box
Power Rule: $d/dx [x^n] = nx^{n-1}$Quotient Rule: $d/dx [u/v] = \frac{v \cdot u' - u \cdot v'}{v^2}$Trig: $d/dx [\sin x] = \cos x$; $d/dx [\cos x] = -\sin x$
Dimensional Analysis of Derivatives
The derivative $dy/dx$ always has dimensions of $[Y]/[X]$. If $y$ is displacement $[L]$ and $x$ is time $[T]$, then $dy/dx$ is $[LT^{-1}]$ (Velocity). If you calculate a second derivative $d^2y/dx^2$, the dimensions are $[Y]/[X^2]$, which is $[LT^{-2}]$ (Acceleration). Use this to verify your Physics derivations!
Variations: Implicit Differentiation
When $x$ and $y$ are tangled (e.g., $x^2 + y^2 = 25$), we differentiate every term with respect to $x$, treating $y$ as a function of $x$ (adding a $dy/dx$ term whenever we differentiate a $y$). This is essential for Related Rates problems (e.g., how fast the water level rises in a cone).
Shortcuts & Mnemonics
- The "C" Rule: In Trigonometry, the derivatives of all functions starting with "C" ($\cos, \cot, \text{cosec}$) are negative.
- High-Low Rule: For the Quotient Rule: "Low D-High minus High D-Low, over the square of what's below."
- $e^x$ stays $e^x$: The most "loyal" function in calculus; it is its own derivative.
Edge Cases
- $|x|$ at $x=0$: The function is continuous but not differentiable because there is a "sharp corner" (the slope changes abruptly from $-1$ to $1$).
- Vertical Tangent: If $f'(x) \to \infty$, the derivative is undefined (e.g., $y = x^{1/3}$ at $x=0$).
- Derivative of a Constant: Always zero. Physically, if something isn't changing, its rate of change is null.
