Differentiate y w.r.t. x where y=e^(cot^-1 x^4)^7
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Before solving we must must know this formula
tan⁻¹(x)=cot⁻¹(1/x):
![]y= e^{ ( cot^{-1} x^{4})^{7} }= e^{ ( tan^{-1} 1/x^{4})^{7} } \\ dy/dx =e^{ ( cot^{-1} x^{4})^{7} }* 7(tan^{-1} 1/x^{4})^6* \frac{1}{1+(1/ x^{4})^{2} } *(-4) x^{-5} \\ dy/dx=-28 \frac{e^{ ( cot^{-1} x^{4})^{7} }*(tan^{-1} 1/x^{4})*}{ x^{5} (1+1/ x^{4})}=-28 \frac{e^{ ( cot^{-1} x^{4})^{7} }*(cot^{-1} x^{4})*}{ x^{5} (1+1/ x^{4})}](https://tex.z-dn.net/?f=%5Dy%3D+e%5E%7B+%28+cot%5E%7B-1%7D+x%5E%7B4%7D%29%5E%7B7%7D+%7D%3D+e%5E%7B+%28+tan%5E%7B-1%7D+1%2Fx%5E%7B4%7D%29%5E%7B7%7D+%7D+%5C%5C+dy%2Fdx+%3De%5E%7B+%28+cot%5E%7B-1%7D+x%5E%7B4%7D%29%5E%7B7%7D+%7D%2A+7%28tan%5E%7B-1%7D+1%2Fx%5E%7B4%7D%29%5E6%2A+%5Cfrac%7B1%7D%7B1%2B%281%2F+x%5E%7B4%7D%29%5E%7B2%7D+%7D+%2A%28-4%29+x%5E%7B-5%7D+%5C%5C+dy%2Fdx%3D-28+%5Cfrac%7Be%5E%7B+%28+cot%5E%7B-1%7D+x%5E%7B4%7D%29%5E%7B7%7D+%7D%2A%28tan%5E%7B-1%7D+1%2Fx%5E%7B4%7D%29%2A%7D%7B+x%5E%7B5%7D+%281%2B1%2F+x%5E%7B4%7D%29%7D%3D-28+%5Cfrac%7Be%5E%7B+%28+cot%5E%7B-1%7D+x%5E%7B4%7D%29%5E%7B7%7D+%7D%2A%28cot%5E%7B-1%7D+x%5E%7B4%7D%29%2A%7D%7B+x%5E%7B5%7D+%281%2B1%2F+x%5E%7B4%7D%29%7D "]y= e^{ ( cot^{-1} x^{4})^{7} }= e^{ ( tan^{-1} 1/x^{4})^{7} } \\ dy/dx =e^{ ( cot^{-1} x^{4})^{7} }* 7(tan^{-1} 1/x^{4})^6* \frac{1}{1+(1/ x^{4})^{2} } *(-4) x^{-5} \\ dy/dx=-28 \frac{e^{ ( cot^{-1} x^{4})^{7} }*(tan^{-1} 1/x^{4})*}{ x^{5} (1+1/ x^{4})}=-28 \frac{e^{ ( cot^{-1} x^{4})^{7} }*(cot^{-1} x^{4})*}{ x^{5} (1+1/ x^{4})}")
tan⁻¹(x)=cot⁻¹(1/x):
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