Mad Differentiation

Calculus Level 4

$\large \sin(\cos(\tan(\csc(\sec(\cot(xy))))))=\frac{\pi}{2016}$

If $\dfrac{dy}{dx} = -2016$ and $\dfrac{y}{x}=k$ , find $k$ .

The answer is 2016.

1 solution

Tommy Li
Nov 23, 2016

$\large \sin(\cos(\tan(\csc(\sec(\cot(xy))))))=\frac{\pi}{2016}$

Differentiate both sides with respect to $\large x$ :

$\large (...)(y+x\dfrac{dy}{dx})=0$

$\large (y+x\dfrac{dy}{dx})=0$

$\large \dfrac{dy}{dx}= -\dfrac{y}{x} =-2016$

$\large \dfrac{y}{x} =2016$

$\large \Rightarrow k=2016$

Nice idea. However, you still have to explain why the left factor doesn't yield a zero term.

Calvin Lin Staff - 4 years, 6 months ago

@Calvin Lin Sir , how can we do that?..I couldn't understand how to show that the left factor is non - zero..

Ankit Kumar Jain - 4 years, 1 month ago

Sir..I have done it..But it included a lot of work, I mean casework sort of...Please provide some explanation as to why it can't be $0$ .

Ankit Kumar Jain - 4 years, 1 month ago

Short of doing the actual differentiation, no immediately obvious reason to me why it doesn't yield a zero term.

Calvin Lin Staff - 4 years, 1 month ago

@Calvin Lin – Sir , I tried to do it this way that after differentiation like the first term of the product will be sin converted to cos in the question..So if it has to be $0$ then the angle of it i.e cos(tan(csc(sec(cot(xy))))) = pi/2 implies that sin(cos(tan(csc(sec(cot(xy)))))) = 1 and not equal to pi/2016...We can use a similar analysis for other terms as well.

Ankit Kumar Jain - 4 years, 1 month ago

Mad Differentiation

The answer is 2016.

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