Methods of Differentiation — Class 12 Mathematics PYQs

Consider the following for the next two (02) items that follow:

Let $f(x)$ and $g(x)$ be two functions such that $g(x)=x-\frac{1}{x}$ and $f\circ g(x)=x^3-\frac{1}{x^3}$.

What is $f^{\prime \prime }(x)$ equal to?

Direction : Consider the following for the two (02) items that follow :

Let $f:(-1,1)\to \mathbb{R}$ be a differentiable function with $f(0)=-1$ and $f^{\prime }(0)=1$. Let $h(x)=f(2f(x)+2)$ and $g(x)=(h(x){)}^2$.

What is $h^{\prime }(0)$ equal to?

Consider the following for the next two (02) items that follow :

Let a function f be defined on $\mathbb{R}-\{0\}$ and $2f(x)+f\left(\frac{1}{x}\right)=x+3$.

If $f$ is differentiable, then what is $f^{\prime }(0.5)$ equal to ?

If $f(x)=|\ln |x||$ where $0<x<1$, then what is $f^{\prime }(0.5)$ equal to ?

If $f^{\prime }(x)=\cos (\ln x)$ and $y=f\left(\frac{2x-3}{x}\right)$, then what is $\frac{dy}{dx}$ equal to ?

Consider the following for the next three (03) items that follow :

Let $f(x)=P{e}^x+Q{e}^{2x}+R{e}^{3x}$, where $P,Q,R$ are real numbers. Further $f(0)=6$, $f^{\prime }(\ln 3)=282$ and ${\int }_0^{\ln 2}f(x)dx=11$

What is $f^{\prime }(0)$ equal to?

Consider the following for the next three (03) items that follow :

Let $f(x)=|\ln x|,x\ne 1$

What is the derivative of $f(x)$ at $x=0\cdot 5$?

Consider the following for the next three (03) items that follow :

Let $f(x)=|\ln x|,x\ne 1$

What is the derivative of $f(x)$ at $x=2$?

Consider the following for the next three (03) items that follow :

Let $f(x)=|\ln x|,x\ne 1$

What is the derivative of $f\circ f(x)$, where $1<x<2$?

What is the derivative of ${\sin }^{2}x$ with respect to ${\cos }^{2}x$?