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# Differentiation Rules

These are the rules that explain how to take derivatives of any functions: from polynomials to trigonometric functions to logarithms.

What is the value of the derivative of \(y = \left|\ln(x^2)\right|\) at \(x = -\dfrac{1}{2}\)?

Above shows the derivative of \(\sin(x) \) by the first principle.

What's the derivative of \(\sin(x^{\circ})\)?

If \(\displaystyle\frac{d}{dx} f(x) = g(x)\) and \(\displaystyle\frac{d}{dx}g(x) = f(x^2)\), then \(\displaystyle\frac{d^2}{dx^2}f(x^3) =\ ?\)

\[ \begin{align} (A) &\quad f\left(x^6\right) & (B)&\quad g\left(x^3\right)\\ (C) &\quad 3x^2 g\left(x^3\right) & (D) &\quad 9x^4 f\left(x^6\right)+6x g\left(x^3\right)\\ (E) &\quad f\left(x^6\right) + g\left(x^3\right) & & \end{align} \]

*Credit: 1969 AP Calculus AB Exam*

\[y = f(x), p = \dfrac{dy}{dx}, q = \dfrac{d^2y}{dx^2} \]

What is \( \dfrac{d^2x}{dy^2} \ ? \)

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