Question

Differentiate y w.r.t. x where y=e^(cot^-1 x^4)^7

Answer 1

Before solving we must must know this formula
tan⁻¹(x)=cot⁻¹(1/x):

$]y= e^{ ( cot^{-1} x^{4})^{7} }= e^{ ( tan^{-1} 1/x^{4})^{7} } \\ dy/dx =e^{ ( cot^{-1} x^{4})^{7} }* 7(tan^{-1} 1/x^{4})^6* \frac{1}{1+(1/ x^{4})^{2} } *(-4) x^{-5} \\ dy/dx=-28 \frac{e^{ ( cot^{-1} x^{4})^{7} }*(tan^{-1} 1/x^{4})*}{ x^{5} (1+1/ x^{4})}=-28 \frac{e^{ ( cot^{-1} x^{4})^{7} }*(cot^{-1} x^{4})*}{ x^{5} (1+1/ x^{4})}$