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## Derivatives

A derivative is simply a rate of change. Whether you're modeling the movement of a particle or a supply/demand model, this is a key instrument of Calculus.

# Level 2

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?

If $$|f(x_1)-f(x_2)|=(x_1-x_2)^2$$, then find the tangent to $$y=f(x)$$ at $$(2,5)$$.

If the tangent can be represented as $$y=mx+c$$, find the value of $$m+c$$.

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

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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