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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.

Differentiation Rules: Level 2 Challenges

What is the following derivative equal to? $\dfrac {d}{du} \left[ \dfrac {u^{n+1}}{(n+1)^2} \cdot \left[ (n+1) \ln u - 1 \right] \right].$

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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