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Differentiation Rules

Differentiation Rules: Level 2 Challenges


What is the following derivative equal to? ddu[un+1(n+1)2[(n+1)lnu1]].\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=ln(x2)y = \left|\ln(x^2)\right| at x=12x = -\dfrac{1}{2}?

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

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

If ddxf(x)=g(x)\displaystyle\frac{d}{dx} f(x) = g(x) and ddxg(x)=f(x2)\displaystyle\frac{d}{dx}g(x) = f(x^2), then d2dx2f(x3)= ?\displaystyle\frac{d^2}{dx^2}f(x^3) =\ ?

(A)f(x6)(B)g(x3)(C)3x2g(x3)(D)9x4f(x6)+6xg(x3)(E)f(x6)+g(x3) \begin{aligned} (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{aligned}

Credit: 1969 AP Calculus AB Exam

y=f(x),p=dydx,q=d2ydx2y = f(x), p = \dfrac{dy}{dx}, q = \dfrac{d^2y}{dx^2}

What is d2xdy2 ? \dfrac{d^2x}{dy^2} \ ?


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