Questions in differentiation

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If $f(x)$ is a differentiable function and ${f}''(0)=a$ then $\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}$ is
If $x=A\cos 4t+B\sin 4t$,then $\frac{{{d}^{2}}x}{d{{t}^{2}}}=$
If $f(x)={{\tan }^{-1}}\left\{ \frac{\log \left( \frac{e}{{{x}^{2}}} \right)}{\log (e{{x}^{2}})} \right\}+{{\tan }^{-1}}\left( \frac{3+2\log x}{1-6\log x} \right)$, then $\frac{{{d}^{n}}y}{d{{x}^{n}}}$ is $(n\ge 1)$
If ${{f}_{n}}(x)$, ${{g}_{n}}(x)$, ${{h}_{n}}(x),n=1,\,2,\,3$are polynomials in x such that ${{f}_{n}}(a)={{g}_{n}}(a)={{h}_{n}}(a),n=1,2,3$ and $F(x) = \begin{vmatrix} {{f_1}(x)}&{{f_2}(x)}&{{f_3}(x)}\\ {{g_1}(x)}&{{g_2}(x)}&{{g_3}(x)}\\ {{h_1}(x)}&{{h_2}(x)}&{{h_3}(x)} \end{vmatrix}$. Then ${F}'(a)$is equal to
Let $f(x) = \begin{vmatrix} {{x^3}}&{\sin x}&{\cos x}\\ 6&{ - 1}&0\\ p&{{p^2}}&{{p^3}} \end{vmatrix}$, where $p$ is a constant. Then $\frac{{{d}^{3}}}{d{{x}^{3}}}\left\{ f(x) \right\}$at $x=0$is
$f(x) = \begin{vmatrix} {{x^3}}&{{x^2}}&{3{x^2}}\\ 1&{ - 6}&4\\ p&{{p^2}}&{{p^3}} \end{vmatrix}$, here $p$ is a constant, then $\frac{{{d}^{3}}f(x)}{d{{x}^{3}}}$ is
If $y=\sin px$ and ${{y}_{n}}$ is the $n^{th}$ derivative of $y$, then $\begin{vmatrix} y&{{y_1}}&{{y_2}}\\ {{y_3}}&{{y_4}}&{{y_5}}\\ {{y_6}}&{{y_7}}&{{y_8}} \end{vmatrix}$ is equal to
If ${{y}^{2}}=a{{x}^{2}}+bx+c$, then ${{y}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}}$is
If $y={{a}^{x}}.{{b}^{2x-1}}$, then $\frac{{{d}^{2}}y}{d{{x}^{2}}}$ is
If $z=\frac{{{({{x}^{4}}+{{y}^{4}})}^{1/3}}}{{{({{x}^{3}}+{{y}^{3}})}^{1/4}}}$, then $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=$

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