Differentiation

The value of $\log_e2\,\dfrac{d}{dx}\left(\log_{\cos x}\operatorname{cosec}x\right)$ at $x=\dfrac\pi4$ is

Let M and N be the number of points on the curve $y^5-9xy+2x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M+N$ equals ______.

For the curve $C:\left(x^2+y^2-3\right)+\left(x^2-y^2-1\right)^5=0$, the value of $3y'-y^3y''$, at the point $(\alpha,\alpha)$, $\alpha>0$, on C, is equal to ______.

Let $x(t)=2\sqrt2\cos t\sqrt{\sin2t}$ and $y(t)=2\sqrt2\sin t\sqrt{\sin2t}$, $t\in\left(0,\dfrac\pi2\right)$. Then $\dfrac{1+\left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}}$ at $t=\dfrac\pi4$ is equal to :

If $y(x)=\tan x-\displaystyle\int_0^{2x}(x^2-xt+1)\sin t\,dt$, then at $x=\dfrac\pi3$, the value of $2y'''-y''$ is equal to

The value of $6m_2 + 4n_2 + 8m_2n_2$ is ____.

For any positive integer $n$, define $f_n: (0, \infty) \to \mathbb{R}$ as

$$f_n(x) = \sum_{j=1}^{n} \tan^{-1}\left(\frac{1}{1 + (x+j)(x+j-1)}\right) \quad \text{for all } x \in (0, \infty).$$

(Here, the inverse trigonometric function $\tan^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.)

Then, which of the following statement(s) is (are) TRUE?

If for $x\in\left(0,\dfrac14\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt x}{1-9x^3}\right)$ is $\sqrt x\cdot g(x)$, then $g(x)$ equals :

The tangent at the point (2, −2) to the curve, $x^2y^2-2x=4(1-y)$ does not pass through the point :

If $y=\left[x+\sqrt{x^2-1}\right]^{15}+\left[x-\sqrt{x^2-1}\right]^{15}$, then $(x^2-1)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}$ is equal to :

If $2x=y^{1/5}+y^{-1/5}$ and $(x^2-1)\dfrac{d^2y}{dx^2}+\lambda x\dfrac{dy}{dx}+ky=0$, then λ + k is equal to :

Let $f$ be a polynomial function such that $f(3x)=f'(x)\cdot f''(x)$, for all $x\in\mathbb R$. Then :

Let $f:\mathbb R\to\mathbb R$, $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ be differentiable functions such that $f(x)=x^3+3x+2$, $g(f(x))=x$ and $h(g(g(x)))=x$ for all $x\in\mathbb R$. Then

For $x\in\mathbb R$, $f(x)=|\log2-\sin x|$ and $g(x)=f(f(x))$, then :

If the tangent at a point P, with parameter t, on the curve $x=4t^2+3$, $y=8t^3-1$, $t\in\mathbb R$, meets the curve again at a point Q, then the coordinates of Q are :

Consider the hyperbola $H:x^2-y^2=1$ and a circle $S$ with center $N(x_2,0)$. Suppose that $H$ and $S$ touch each other at a point $P(x_1,y_1)$ with $x_1>1$ and $y_1>0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If $(l,m)$ is the centroid of the triangle $\Delta PMN$, then the correct expression(s) is(are)

The correct statement(s) is(are)

The distance, from the origin, of the normal to the curve, $x=2\cos t+2t\sin t$, $y=2\sin t-2t\cos t$ at $t=\dfrac\pi4$, is :

The equation of a normal to the curve, $\sin y=x\sin\left(\dfrac\pi3+y\right)$ at $x=0$, is :

The slope of the tangent to the curve $(y-x^5)^2=x(1+x^2)^2$ at the point $(1,3)$ is