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# Puzzles

## Two tangents

Source: Reddit
Find a line which is tangent to the curve $$y=x^4-4x^3$$ at 2 points.

## Double derivative

What is
$$\frac{d}{dy}\left(\frac{dy}{dx}\right)$$
when:
(i) $$y=x$$
(ii) $$y=x^2$$
(iii) $$y=x^3$$
(iv) $$y=x^n$$
(v) $$y=e^x$$
(vi) $$y=\sin(x)$$?

## Differentiate this

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$
Find $$f'(x)$$.

## x to the power of x again

Let $$y=x^{x^{x^{x^{...}}}}$$ [$$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ...))) with an infinite number of $$x$$s]. What is $$\frac{dy}{dx}$$?

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