$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 3{\cos\left({x}\right)}^{4} \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {\cot\left({x}\right)}^{2}-{\cos\left({x}\right)}^{2} \right) }\right)$
Find the derivative$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ -12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {\cot\left({x}\right)}^{2}-{\cos\left({x}\right)}^{2} \right) }\right)$
Find the derivative$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ -12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ -\frac{ 2\cos\left({x}\right) }{ {\sin\left({x}\right)}^{3} }+\sin\left({2x}\right) }\right)$
Write all numerators above the common denominator$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ -12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ \frac{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3}\sin\left({2x}\right) }{ {\sin\left({x}\right)}^{3} } }\right)$
Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ \frac{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3}\sin\left({2x}\right) }{ {\sin\left({x}\right)}^{3} } }\right)$
Simplify the complex fraction$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3}\sin\left({2x}\right) }\right)$
Use $\sin\left({2t}\right)=2\sin\left({t}\right)\cos\left({t}\right)$ to expand the expression$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3} \times 2\sin\left({x}\right)\cos\left({x}\right) }\right)$
Factor out $-2\cos\left({x}\right)$ from the expression$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ -2\cos\left({x}\right) \times \left( 1-{\sin\left({x}\right)}^{3}\sin\left({x}\right) \right) }\right)$
Cancel out the common factor $-2$$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ \cos\left({x}\right) \times \left( 1-{\sin\left({x}\right)}^{3}\sin\left({x}\right) \right) }\right)$
Cancel out the common factor $\cos\left({x}\right)$$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ 1-{\sin\left({x}\right)}^{3}\sin\left({x}\right) }\right)$
Calculate the product$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ 1-{\sin\left({x}\right)}^{4} }\right)$
Use ${a}^{2}-{b}^{2}=\left( a-b \right)\left( a+b \right)$ to factor the expression$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ \left( 1-{\sin\left({x}\right)}^{2} \right) \times \left( 1+{\sin\left({x}\right)}^{2} \right) }\right)$
Use $1-{\sin\left({t}\right)}^{2}={\cos\left({t}\right)}^{2}$ to simplify the expression$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ {\cos\left({x}\right)}^{2} \times \left( 1+{\sin\left({x}\right)}^{2} \right) }\right)$
Cancel out the common factor ${\cos\left({x}\right)}^{2}$$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\sin\left({x}\right)}^{4} }{ 1+{\sin\left({x}\right)}^{2} }\right)$
Determine the sign of the fraction$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ 6{\sin\left({x}\right)}^{4} }{ 1+{\sin\left({x}\right)}^{2} }\right)$
Evaluate the limit$\frac{ 6{\sin\left({\frac{ π }{ 2 }}\right)}^{4} }{ 1+{\sin\left({\frac{ π }{ 2 }}\right)}^{2} }$
Simplify the expression$3$