Differentiation of Determinants

$If\text{f}\left(x\right)=\left|\begin{array}{ccc}\text{cos}\left(x+x^2\right)& sin\left(x+x^2\right)& -cos\left(x+x^2\right)\\ sin\left(x-x^2\right)& cos\left(x-x^2\right)& sin\left(x-x^2\right)\\ sin2x& 0& sin2x^2\end{array}\right|then,\; find { f}^{\prime }\left(x\right)$

$If\text{f}\left(x\right)=\left|\begin{array}{ccc}{\left(x-a\right)}^4& {\left(x-a\right)}^3& 1\\ {\left(x-b\right)}^4& {\left(x-b\right)}^3& 1\\ {\left(x-c\right)}^4& {\left(x-c\right)}^3& 1\end{array}\right|then{\text{f}}^{\prime }\left(x\right)=\lambda ·\left|\begin{array}{ccc}{\left(x-a\right)}^4& {\left(x-a\right)}^2& 1\\ {\left(x-b\right)}^4& {\left(x-b\right)}^2& 1\\ {\left(x-c\right)}^4& {\left(x-c\right)}^2& 1\end{array}\right|·\text{Find the value of}\lambda$

Let $\mathrm{ƒ}\left(\mathrm{x}\right)=\left|\begin{array}{lll}{\mathrm{x}}^3& \sin \mathrm{x}& \cos \mathrm{x}\\ 6& -1& 0\\ \mathrm{p}& {\mathrm{p}}^2& {\mathrm{p}}^3\end{array}\mathrm{ }\right|$ , where $\mathrm{p}$ is a constant. Then $\frac{{\mathrm{d}}^3}{\mathrm{d}{\mathrm{x}}^3}\left\{\mathrm{ƒ}\left(\mathrm{x}\right)\right\}$ at $\mathrm{x}=0$ is -

Find the derivative of $\left|\begin{array}{ll}{a}^x& 1\\ e^x& 4\end{array}\right|$.

Find the derivative of $\left|\begin{array}{ll}\log x& 1\\ e^x& 4\end{array}\right|$.

Find the derivative of $\left|\begin{array}{ll}\sin x& 1\\ \cos x& 4\end{array}\right|$.

Find the derivative of $\left|\begin{array}{ll}{x}^2& 1\\ 2x& 4\end{array}\right|$.

If $\mathrm{f}\text{(}\mathrm{x}\text{)}=\left|\begin{array}{lll}(1+\mathrm{x}{)}^{{\mathrm{a}}_1}& (1+\mathrm{x}{)}^{{\mathrm{a}}_2}& (1+\mathrm{x}{)}^{{\mathrm{a}}_3}\\ (1+\mathrm{x}{)}^{{\mathrm{b}}_1}& (1+\mathrm{x}{)}^{{\mathrm{b}}_2}& (1+\mathrm{x}{)}^{{\mathrm{b}}_3}\\ (1+\mathrm{x}{)}^{{\mathrm{c}}_1}& (1+\mathrm{x}{)}^{{\mathrm{c}}_2}& (1+\mathrm{x}{)}^{{\mathrm{c}}_3}\end{array}\right|$ $+3{\mathrm{x}}^2+19\mathrm{x}+6,$ then the coefficient of $\mathrm{x}$ in the expansion of $\mathrm{f}\left(\mathrm{x}\right)$ is :

If $f\left(x\right)=\left|\begin{array}{ccc}{x}^3& 2x^2+1& 1+3x\\ 3x^2+2& 2x& x^3+6\\ x^3-x& 4& x^2-2\end{array}\right|$ for all $x\in ℝ$, then $2f\left(0\right)+f^{\text{'}}\left(0\right)$ is equal to

If $f\left(x\right)=\left|\begin{array}{ccc}{x}^3& 2x^2+1& 1+3x\\ 3x^2+2& 2x& x^3+6\\ x^3-x& 4& x^2-2\end{array}\right|$ for all $x\in R$, then  $2f\left(0\right)+f^{\text{'}}\left(0\right)$ is equal to

If ${\mathrm{\Delta }}_1=\left|\begin{array}{ccc}x& a& b\\ b& x& a\\ a& b& x\end{array}\right|$ and ${\mathrm{\Delta }}_2=\left|\begin{array}{cc}x& b\\ a& x\end{array}\right|$ are the given determinants, then

Let $∆\left(x\right)=\left|\begin{array}{ccc}{\cos }^{2}x& \cos x\sin x& -\sin x\\ \cos x\sin x& {\sin }^{2}x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$ then $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left[∆\left(x\right)+{∆}^{\prime }\left(x\right)\right]dx$ equals

If $f\left(x\right), g\left(x\right)$ and $h\left(x\right)$ are three polynomials of degree $3$ then $\varnothing \left(x\right)=\left|\begin{array}{ccc}{f}^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\\ f^{\prime \prime \prime }\left(x\right)& g^{\prime \prime \prime }\left(x\right)& h^{\prime \prime \prime }(x) \end{array}\right|$is a polynomial of degree

If ${∆}_1=\left|\begin{array}{ccc}x& b& b\\ a& x& b\\ a& a& x\end{array}\right|$ and ${∆}_2=\left|\begin{array}{cc}x& b\\ a& x\end{array}\right|,$ then

If $f\left(x\right)= \left|\begin{array}{ccc}1& 2a& 3a^2\\ x& x^2& x^3\\ e^{x-a}& e^{x^2-a^2}& e^{x^3-a^3}\end{array}\right|$ then $f^{\prime }\left(a\right)=$

Let $\mathrm{ƒ}\left(\mathrm{x}\right)=\left|\begin{array}{lll}{\mathrm{x}}^3& \sin \mathrm{x}& \cos \mathrm{x}\\ 6& -1& 0\\ \mathrm{p}& {\mathrm{p}}^2& {\mathrm{p}}^3\end{array}\mathrm{ }\right|$ , where $\mathrm{p}$ is a constant. Then $\frac{{\mathrm{d}}^3}{\mathrm{d}{\mathrm{x}}^3}\left\{\mathrm{ƒ}\left(\mathrm{x}\right)\right\}$ at $\mathrm{x}=0$ is -

Let $\mathrm{ƒ}\left(\mathrm{x}\right)=\left|\begin{array}{lll}{\mathrm{x}}^3& \sin \mathrm{x}& \cos \mathrm{x}\\ 6& -1& 0\\ \mathrm{p}& {\mathrm{p}}^2& {\mathrm{p}}^3\end{array}\mathrm{ }\right|$ , where $\mathrm{p}$ is a constant. Then $\frac{{\mathrm{d}}^3}{\mathrm{d}{\mathrm{x}}^3}\left\{\mathrm{ƒ}\left(\mathrm{x}\right)\right\}$ at $\mathrm{x}=0$ is -

The value of $\mathrm{\lambda }$ if ${\mathrm{a}\mathrm{x}}^4\mathrm{ }+\mathrm{ }{\mathrm{b}\mathrm{x}}^3\mathrm{ }+\mathrm{ }{\mathrm{c}\mathrm{x}}^2\mathrm{ }+\mathrm{ }50\mathrm{x}\mathrm{ }+\mathrm{ }\mathrm{d}=\mathbf{ }\left|\mathrm{ }\begin{array}{lll}{\mathrm{x}}^3-14{\mathrm{x}}^2& -\mathrm{x}& 3\mathrm{x}+\mathrm{\lambda }\\ 4\mathrm{x}+1& 3\mathrm{x}& \mathrm{x}-4\\ -3& 4& 0\end{array}\mathrm{ }\right|$ is

If $\mathrm{\alpha }$ be a repeated roots of the quadratic equation $\mathrm{f}\left(\mathrm{x}\right)=0$ and $\mathrm{A}\left(\mathrm{x}\right),\mathrm{ }\mathrm{B}\left(\mathrm{x}\right),\mathrm{ }\mathrm{C}\left(\mathrm{x}\right)$ are polynomials of degree 3, 4, 5 respectively then determinant $\left|\mathrm{ }\begin{array}{lll}\mathrm{A}\left(\mathrm{x}\right)& \mathrm{B}\left(\mathrm{x}\right)& \mathrm{C}\left(\mathrm{x}\right)\\ \mathrm{A}\left(\mathrm{\alpha }\right)& \mathrm{B}\left(\mathrm{\alpha }\right)& \mathrm{C}\left(\mathrm{\alpha }\right)\\ \mathrm{A}\mathrm{\prime }\left(\mathrm{\alpha }\right)& \mathrm{B}\mathrm{\prime }\left(\mathrm{\alpha }\right)& \mathrm{C}\mathrm{\prime }\left(\mathrm{\alpha }\right)\end{array}\mathrm{ }\right|$ is divisible by (where $\mathrm{A}\mathrm{\prime }\left(\mathrm{\alpha }\right)\mathrm{ }={\left(\frac{\mathrm{d}\mathrm{A}}{\mathrm{d}\mathrm{x}}\right)}_{\mathrm{x}=\mathrm{\alpha }}$ , etc) -

If $∆\left(x\right)=\left|\begin{array}{ccc}{x}^n& \sin x& \cos x\\ n!& \sin \frac{n\mathrm{\pi }}{2}& \cos \frac{n\mathrm{\pi }}{2}\\ a& a^2& a^3\end{array}\right|,$ then the value of $\frac{d^n}{d{x}^n}\left[∆\left(x\right)\right]$ at $x=0$ is