Differentiation · Mathematics · COMEDK

$$ \text { The second derivative of } \sin 3 \boldsymbol{x} \boldsymbol{\operatorname { c o s }} \mathbf{5 x} \text { is: } $$

$$ \text { If } y=\tan ^{-1}\left(\frac{\sqrt{1+x^3}+\sqrt{1-x^3}}{\sqrt{1+x^3}-\sqrt{1-x^3}}\right) \text { then } \frac{\boldsymbol{d} \boldsymbol{y}}{\boldsymbol{d x}}= $$

If $\log y=\log (\sin x)-x^2$, then $\frac{d^2 y}{d x^2}+\mathbf{4} x \frac{d y}{d x}+\mathbf{4} x^2 y=$

$$ \text { If } x=a(\theta-\sin \theta) \text { and } y=a(1-\cos \theta) \text {, then } \frac{\left(\mathbf{1}+\boldsymbol{y}_{\mathbf{1}}{ }^{\mathbf{2}}\right)^{\mathbf{3} / \mathbf{2}}}{\boldsymbol{y}_{\mathbf{2}}}= $$

The derivative of $y=\sin ^2\left[\cot ^{-1}\left(\sqrt{\frac{\mathbf{1}-\boldsymbol{x}}{\mathbf{1}+\boldsymbol{x}}}\right)\right]$ is

If $y=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$,

then $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}=$

$$ \text { If } y=f(x), \quad p=\frac{d y}{d x} ; q=\frac{d^2 y}{d x^2} \text { then } \frac{d^2 x}{d y^2} \text { is equal to } $$

$$ \text { If } y=\sqrt{\sin x+y} \text { then find } \frac{d y}{d x} \text { at } x=0, \quad y=1 $$

$$ \text { If } y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^2}}{13}\right) \text { then } \frac{d y}{d x} \text { equals } $$

$$ \text { If } x^2+y^2=t+\frac{1}{t} \text { and } x^4+y^4=t^2+\frac{1}{t^2} \text { then } \frac{d y}{d x}= $$

If $$f(x)=f^{\prime}(x)$$ and $$f(1)=2$$, then $$f(3)$$ is

$$ \text { If } y=\log _e\left(\frac{x^2}{e^2}\right) \text {, then } \frac{d^2 y}{d x^2} \text { is equal to } $$

$$ \text { If } y=\tan ^{-1}\left(\frac{3-2 x}{1+6 x}\right) \text { then } \frac{d y}{d x} \text { is } $$

$$ \text { If } \sin y=x(\cos (a+y)) \text {, then find } \frac{d y}{d x} \text { when } x=0 $$

The approximate value of $$f(5.001)$$, where $$f(x)=x^3-7 x^2+10$$

$$ \text { If } f(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right) \text { then } f^{\prime}(0) \text { is equal to } $$

If $$f(x)=\frac{(x+1)^7 \sqrt{1+x^2}}{\left(x^2-x+1\right)^6}$$ then the value of $$f^{\prime}(0)$$ is equal to

The equation of normal to the curve $$y = {(1 + x)^y} + {\sin ^{ - 1}}({\sin ^2}x)$$ at $$x = 0$$ is

If $$y = {2^{\log x}}$$, then $${{dy} \over {dx}}$$ is

If $$y = {\cos ^2}{{3x} \over 2} - {\sin ^2}{{3x} \over 2}$$, then $${{{d^2}y} \over {d{x^2}}}$$ is

If $${x^x} = {y^y}$$, then $${{dy} \over {dx}}$$ is