Let f(x)=det(exsin(x)x2ln(x))f(x) = \det \begin{pmatrix} e^x & \sin(x) \\ x^2 & \ln(x) \end{pmatrix}f(x)=det(exx2sin(x)ln(x)). Find f′(x)f'(x)f′(x).
exln(x)+exx−2xsin(x)−x2cos(x)e^x \ln(x) + \frac{e^x}{x} - 2x \sin(x) - x^2 \cos(x)exln(x)+xex−2xsin(x)−x2cos(x)
exln(x)+exx−(2xsin(x)+x2cos(x))e^x \ln(x) + \frac{e^x}{x} - (2x \sin(x) + x^2 \cos(x))exln(x)+xex−(2xsin(x)+x2cos(x))
exln(x)+exx−1−2xsin(x)+x2cos(x)e^x \ln(x) + e^x x^{-1} - 2x \sin(x) + x^2 \cos(x)exln(x)+exx−1−2xsin(x)+x2cos(x)
exln(x)−exx−2xsin(x)−x2cos(x)e^x \ln(x) - \frac{e^x}{x} - 2x \sin(x) - x^2 \cos(x)exln(x)−xex−2xsin(x)−x2cos(x)