Differentiation

Let $\mathbb R$ denote the set of all real numbers. Consider the polynomial function $f:\mathbb R\to\mathbb R$ defined by
$$f(x)=\frac{d^{10}}{dx^{10}}\left(\left(x^2-1\right)^{10}\right),\qquad\text{for all }x\in\mathbb R.$$
Here $\dfrac{d^{10}}{dx^{10}}\left(\left(x^2-1\right)^{10}\right)$ is the $10^{\text{th}}$ order derivative of the function $\left(x^2-1\right)^{10}$.

Then which of the following statements is (are) TRUE ?

If $y=\tan^{-1}\left(\dfrac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2\tan^{-1}\left(\dfrac x{1+\sqrt{1-x^2}}\right)$, then $\dfrac{dy}{dx}$ at $x=\dfrac{\sqrt3}2$ is equal to:

Let $f$ be a real polynomial of degree $n$ such that $f(x)=f'(x)f''(x)$, for all $x\in\mathbb R$. If $f(0)=0$, then $36\left(f'(2)+f''(2)+\displaystyle\int_0^2f(x)\,dx\right)$ is equal to:

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x)=g''(x)$ for all $x\in\mathbf R$, $f'(1)=2g'(1)=4$ and $g(2)=3f(2)=9$. Then $f(25)-g(25)$ is equal to :

Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define

$$g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}$$

for all real numbers $x$.

Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0, 4)$ be functions defined by

$$f(x) = \log_e\left(x^2 + 2x + 4\right),\quad\text{and}\quad g(x) = \frac{4}{1 + e^{-2x}}.$$

Define the composite function $f\circ g^{-1}$ by $(f\circ g^{-1})(x) = f\left(g^{-1}(x)\right)$, where $g^{-1}$ is the inverse of the function $g$.

Then the value of the derivative of the composite function $f\circ g^{-1}$ at $x = 2$ is _____.

If $y=\tan^{-1}\left(\sec x^3-\tan x^3\right)$, $\dfrac\pi2<x^3<\dfrac{3\pi}2$, then

Let $f:\mathbb R\to\mathbb R$ be defined as
$$f(x)=x^3+x-5$$
If $g(x)$ is a function such that $f(g(x))=x$, $\forall\ x\in\mathbb R$, then $g'(63)$ is equal to ______.

If the angle made by the tangent at the point $(x_0,y_0)$ on the curve $x=12(t+\sin t\cos t)$, $y=12(1+\sin t)^2$, $0<t<\dfrac\pi2$, with the positive $x$-axis is $\dfrac\pi3$, then $y_0$ is equal to:

Let $f,g:\mathbb R\to\mathbb R$ be two real valued functions defined as $f(x)=\begin{cases}-|x+3|, & x<0\\e^x, & x\ge0\end{cases}$ and $g(x)=\begin{cases}x^2+k_1x, & x<0\\4x+k_2, & x\ge0\end{cases}$, where $k_1$ and $k_2$ are real constants. If $(g\circ f)$ is differentiable at $x=0$, then $(g\circ f)(-4)+(g\circ f)(4)$ is equal to :

Let $f:\mathbb R\to\mathbb R$ satisfy $f(x+y)=2^xf(y)+4^yf(x)$, $\forall x,y\in\mathbb R$. If $f(2)=3$, then $14\cdot\dfrac{f'(4)}{f'(2)}$ is equal to ______.

If $\cos^{-1}\left(\dfrac y2\right)=\log_e\left(\dfrac x5\right)^5$, $|y|<2$, then :

If $\displaystyle\int\frac{\left(x^2+1\right)e^x}{(x+1)^2}dx=f(x)e^x+C$, where C is a constant, then $\dfrac{d^3f}{dx^3}$ at $x=1$ is equal to :

Let $f(x)=\begin{vmatrix}a&-1&0\\ax&a&-1\\ax^2&ax&a\end{vmatrix}$, $a\in\mathbb R$. Then the sum of the squares of all the values of $a$, for which $2f'(10)-f'(5)+100=0$, is

Let $f$ be a differentiable function in $\left(0,\dfrac\pi2\right)$. If $\displaystyle\int_{\cos x}^1t^2f(t)\,dt=\sin^3x+\cos x$, then $\dfrac1{\sqrt3}f'\left(\dfrac1{\sqrt3}\right)$ is equal to

If $y(x)=\left(x^x\right)^x$, $x>0$, then $\dfrac{d^2x}{dy^2}+20$ at $x=1$ is equal to ______.

Let for $f(x)=a_0x^2+a_1x+a_2$, $f'(0)=1$ and $f'(1)=0$. If $a_0,a_1,a_2$ are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then $f(4)$ is equal to ______.

Let $f$ be a twice differentiable function on $\mathbb R$. If $f'(0)=4$ and $f(x)+\displaystyle\int_0^x(x-t)f'(t)\,dt=\left(e^{2x}+e^{-2x}\right)\cos2x+\frac2ax$, then $(2a+1)^5a^2$ is equal to ______.

Let the area enclosed by the $x$-axis, and the tangent and normal drawn to the curve $4x^3-3xy^2+6x^2-5xy-8y^2+9x+14=0$ at the point $(-2,3)$ be A. Then 8A is equal to ______.

If $a=\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\frac{2n}{n^2+k^2}$ and $f(x)=\sqrt{\dfrac{1-\cos x}{1+\cos x}}$, $x\in(0,1)$, then :