If $x=t^2$ and $y=t^3$, then $\frac{d^2 y}{d x^2}$ is equal to
The value of $c$ in Rolle's theorem for the function $f(x)=x^3-3 x$ in the interval $[0, \sqrt{3}]$ is
For the function $f(x)=x+\frac{1}{x}, x \in[1,3]$, the value of $c$ for mean value theorem is
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ............. .
$|x|+|x-1|$ is continuous everywhere but fails to be differentiable exactly at two points $x=0$ and $x=1$.
So, there can be more such examples of functions.
Derivative of $x^2$ w.r.t. $x^3$ is ............. .
Derivative of $x^2$ w.r.t. $x^3$ is $\frac{2}{3 x}$.
Let $$u=x^2 \text { and } v=x^3$$
$$\begin{array}{ll} \therefore & \frac{d u}{d x}=2 x \text { and } \frac{d v}{d x}=3 x^2 \\ \Rightarrow & \frac{d u}{d v}=\frac{2 x}{3 x^2}=\frac{2}{3 x} \end{array}$$