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 \(\dfrac d{dx} x^n = n x^{n-1} \).

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

\[ \large y_n (x) = e^x \times e^{x^2} \times e^{x^3} \times \ldots \times e^{x^n} \]

For some positive integer \(n\), let \(y_n(x) \) denote function of \(x\) as stated above.

What is the value of the limit below?

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

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

\[\large\displaystyle\lim_{(x,y,z) \to (0,0,0)} \dfrac{e^{-x^{2} - y^{2} - z^{2}} - 1}{x^{2} + y^{2} + z^{2}} =\, ? \]


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