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Derivatives: Level 2 Challenges


Ski lifts are an important cable transport that helps to carry skiers up a hill, so that they can easily enjoy skiing down the hill.

Which portion of the ski lift is the steepest?

We know that ddxxn=nxn1\dfrac d{dx} x^n = n x^{n-1} .

Is it also true that ddxnx=xnx1\dfrac d{dx} n^x = x n^{x-1} ?

yn(x)=ex×ex2×ex3××exn \large y_n (x) = e^x \times e^{x^2} \times e^{x^3} \times \ldots \times e^{x^n}

For some positive integer nn, let yn(x)y_n(x) denote function of xx as stated above.

What is the value of the limit below?

limnddx(yn(x))x=12 \large \lim_{n\to\infty} \left . \frac d{dx} \left ( y_n (x) \right ) \right |_{x=\frac12}

Let a>1 a \gt 1 be the unique real number such that the equation ax=xa^{x} = x has a unique solution. Compute 1000a\lceil 1000a \rceil.

lim(x,y,z)(0,0,0)ex2y2z21x2+y2+z2=?\large\displaystyle\lim_{(x,y,z) \to (0,0,0)} \dfrac{e^{-x^{2} - y^{2} - z^{2}} - 1}{x^{2} + y^{2} + z^{2}} =\, ?


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