Differentiation Notation and Basic Rules
📌 Differentiation and derivatives are interchangeable terms, represented by notations like $y'$, $\frac{dy}{dx}$, $f'$, or $g'(x)$.
📌 Rule 1 (Constant Rule): The derivative of a constant $c$ is zero ($\frac{d}{dx}(c) = 0$).
📌 Rule 2 (Power Rule): The derivative of $x^n$ is $nx^{n-1}$. If a coefficient exists, it is multiplied by the power (e.g., $\frac{d}{dx}(3x^5) = 3 \cdot 5x^{5-1} = 15x^4$).

Advanced Differentiation Rules
📌 Rule 3 (Square Root Rule): For $\sqrt{u}$, the derivative is $\frac{1}{2\sqrt{u}} \cdot \frac{du}{dx}$, where $\frac{du}{dx}$ is the derivative of the expression inside the root.
📌 Rule 4 (Sum/Difference Rule): The derivative of a sum or difference of functions is the sum or difference of their individual derivatives (e.g., $\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$).
📌 Rule 6 (Product Rule): For the product of two functions, $f(x)g(x)$, the derivative is $f(x)g'(x) + g(x)f'(x)$. Order matters less for multiplication and addition, but it is crucial for division.
📌 Quotient Rule (Division Rule): For $\frac{f(x)}{g(x)}$, the derivative is $\frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$. Always start by squaring the denominator and writing it down first; ensure the order of subtraction is correct (denominator $\times$ derivative of numerator minus numerator $\times$ derivative of denominator).

Chain Rule and Logarithmic Differentiation
📌 Bracket with Power Rule (Chain Rule Application): For $[u]^n$, the derivative is $n[u]^{n-1} \cdot u'$. Always remember to multiply by the derivative of the inside ($u'$).
📌 Rule 9 (Natural Logarithm Rule): The derivative of $\ln(u)$ is $\frac{1}{u} \cdot \frac{du}{dx}$ (i.e., derivative of what is inside goes on top, and what is inside goes in the denominator).
📌 Logarithm Property for Simplification: $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$. This property can simplify expressions before differentiation, but $\ln(b-a)$ cannot be simplified this way.

Trigonometric Derivatives
📌 Basic derivatives are $\frac{d}{dx}(\sin(u)) = \cos(u) \cdot u'$ and $\frac{d}{dx}(\cos(u)) = -\sin(u) \cdot u'$.
📌 The derivative of $\tan(u)$ has two forms: $\frac{1}{\cos^2(u)} \cdot u'$ or $(1 + \tan^2(u)) \cdot u'$.
📌 For all trigonometric functions, always multiply the result by the derivative of the angle inside ($u'$).

Key Points & Insights
➡️ Always simplify the final answer after applying derivative rules, as multiple-choice options are usually in the simplest form.
➡️ When using the Quotient Rule, rigorously follow the sequence: $\frac{\text{Down} \cdot \text{Derivative Up} - \text{Up} \cdot \text{Derivative Down}}{\text{Down}^2}$.
➡️ For composite functions involving brackets with powers or trigonometric functions, the derivative of the inside function ($u'$) must always be applied as a multiplication factor.
➡️ Logarithm properties like $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$ can be used strategically to convert division into subtraction, making individual derivative calculations simpler.

📸 Video summarized with SummaryTube.com on Dec 01, 2025, 17:24 UTC