Question

Coefficient of $x$ in $f(x)= \left|\begin{array}{ccc}x&(1+\sin x)^3&\cos x\\1&\ln (1+x)&2\\x^{2}&(1+x)^{2}&0\end{array}\right|$ is?

Answer 1

Determinant value is to be found by the expansion using a column or a row.  We pick the 1st column as those expressions are complicated.

Determinant D
   = x [ 0 - 2 (1+x)² ] - 1 [ 0 - (1+x)² Cos x ]  + x² [ .... ]
   = - 2 x (1+x)²  + (1+x)² Cos x + x² [ ... ]

Since we want only the coefficient of  x (power 1), we ignore the last term.  We get :
     - 2 x + (1 + 2 x ) Cos x

   It depends whether we can treat Cos x as a coefficient of  x.  We use the MaClaurin (Taylor) series for Cos x to write it as a polynomial in x.
 
 Cos x = 1 - x²/2  + ....     We ignore the x² term, as its power is > 1.       So we replace Cos x by 1.

  Finally,  the coefficient of x :    - 2 + 2  = 0