Differentiation

The value of $g'\left(\frac12\right)$ is

Match List I with List II and select the correct answer using the code given below the lists.

If $g$ is the inverse of a function $f$ and $f'(x)=\dfrac{1}{1+x^5}$, then $g'(x)$ is equal to :

If $y=e^{nx}$, then $\left(\dfrac{d^2y}{dx^2}\right)\left(\dfrac{d^2x}{dy^2}\right)$ is equal to :

Let for $i=1,2,3$, $p_i(x)$ be a polynomial of degree 2 in $x$, $p_i'(x)$ and $p_i''(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let,
$$A(x)=\begin{bmatrix}p_1(x)&p_1'(x)&p_1''(x)\\p_2(x)&p_2'(x)&p_2''(x)\\p_3(x)&p_3'(x)&p_3''(x)\end{bmatrix}$$
and $B(x)=[A(x)]^TA(x)$. Then determinant of $B(x)$ :

Let $f(x)=x|x|$, $g(x)=\sin x$ and $h(x)=(g\circ f)(x)$. Then

For the curve $y=3\sin\theta\cos\theta$, $x=e^\theta\sin\theta$, $0\le\theta\le\pi$, the tangent is parallel to $x$-axis when $\theta$ is :

If $f(x)=x^2-x+5$, $x>\dfrac12$, and $g(x)$ is its inverse function, then $g'(7)$ equals :

Which of the following is true for $0<x<1$?

Let $f(\theta)=\sin\left(\tan^{-1}\left(\dfrac{\sin\theta}{\sqrt{\cos2\theta}}\right)\right)$, where $-\dfrac\pi4<\theta<\dfrac\pi4$. Then the value of
$$\frac{d}{d(\tan\theta)}\left(f(\theta)\right)$$
is

Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x}f(x)=2+\displaystyle\int_0^x\sqrt{t^4+1}\,dt$, for all $x\in(-1,1)$, and let $f^{-1}$ be the inverse function of $f$. Then $(f^{-1})'(2)$ is equal to

If the function $f(x)=x^3+e^{\frac x2}$ and $g(x)=f^{-1}(x)$, then the value of $g'(1)$ is

Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g''(x)$ is continuous, $g(0)\ne0$, $g'(0)=0$, $g''(0)\ne0$, and $f(x)=g(x)\sin x$.

STATEMENT-1: $\displaystyle\lim_{x\to0}\left[g(x)\cot x-g(0)\operatorname{cosec}x\right]=f''(0)$.

and

STATEMENT-2: $f'(0)=g(0)$.

If $f(-10\sqrt2)=2\sqrt2$, then $f''(-10\sqrt2)=$

Let $g(x)=\log f(x)$ where $f(x)$ is a twice differentiable positive function on $(0,\infty)$ such that $f(x+1)=xf(x)$. Then, for $N=1,2,3,\ldots$,
$$g''\left(N+\frac12\right)-g''\left(\frac12\right)=$$

Which of the following is true?

$\dfrac{d^2x}{dy^2}$ equals