Sample QuestionsContinuity and Differentiability questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $y=x \log x$ then value of $\frac{d^2 y}{d x^2}$ :
View full solution →If $3 x+2 y=\sin x$ then $\frac{d y}{d x}$ :
View full solution →If function $f$ is defined such that
$
f(x)=\left\{\begin{array}{cc}
\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\
3, & \text { if } x=\frac{\pi}{2}
\end{array}\right.
$is continuous at $x=\frac{\pi}{2}$, then value of $k$ :
View full solution →If $\left(x^2+y^2\right)^2=x y$, then $\frac{d y}{d x}$ :
View full solution →If $\sin y=x \cos (a+y)$, then $\frac{d x}{d y}$ :
View full solution →Show that function $f(x)=x^2, x=0$ is continuous.
View full solution →Show that function $F ( x )=\frac{1}{(x-a)},$ is discontinuous at $x=a$.
View full solution →If $f(x)=x \sin x$ then find $f^{\prime}\left(\frac{\pi}{2}\right)$.
View full solution →If function $F (x)=\frac{\sin (10 x)}{x}, x \neq 0$, is continuous at $x=0$. Then find the value of $F (0)$.
View full solution →If $x=t^2, y=t^3$ then find $\frac{d^2 y}{d x^2}$.
View full solution →If $y=\sin ^{-1} x$ then find $\frac{d^2 y}{d x^2}$.
View full solution →Find differentiation of $\log (1+\theta)$ w.r.t. $\sin ^{-1} \theta$.
View full solution →If $y=\sqrt{\sin x+y}$, then find $\frac{d y}{d x}$.
View full solution →If $y \sqrt{1-x^2}=\sin ^{-1} x$, then find $\frac{d y}{d x}$
View full solution →If $y=3 \cos x-2 \sin x$, then prove that $\frac{d^2 y}{d x^2}-y=0$
View full solution →Examine the continuity of function.$
f(x)=\left\{\begin{array}{ll}
1+x, & x \leq 3 \\
7-x, & x>3
\end{array} \text { at } x=3\right.
$
View full solution →Differentiation w.r.t. $x$ of $\tan ^{-1}\left[\frac{\sin x+\cos x}{\cos x-\sin x}\right]$.
View full solution →$f(x)=\left\{\begin{array}{c}5 x^2-4, \text { if } x \leq 1 \\ 4 x^2-3 x \text {, if } x>2\end{array}\right.$ Examine the continuity.
View full solution →If $y=x^x+x^a+a^x+a^a$ then find $\frac{d y}{d x}$.
View full solution →If $x^y+y^x=b^a+a^b$ then find $\frac{d y}{d x}$.
View full solution →Find $\frac{d y}{d x}$
$(a)\ \sqrt{x^2+y^2}=\log _e\left(x^2-y^2\right)$
$(b)\ y=x^{x^{x^x} \ldots......\infty}$
$(c)\ x y \log (x+y)=1$
View full solution →Examine the continuity of function at $x=2$ and $x =3 , f(x)=|x-2|+|x-3| $
View full solution →If $x=a \cos \theta+b \sin \theta$ and $y=a \sin \theta-b \cos \theta$ then prove that $y^2 y_2-x y_1+y=0$.
View full solution →If $y=\cos \sqrt{x}$, then the value of $\frac{d y}{d x}$ will be ________ .
View full solution →Dfferentiation of $\cos (\sqrt{x})$ w.r.t. $x$ is ________ .
View full solution →If $3 x+2 y=\cos y$, then $\frac{d y}{d x}$ __________ .
View full solution →If $y=\log _a x$ then $\frac{d y}{d x}=$ __________ .
View full solution →If $y=x+e^x$ then $\frac{d^2 y}{d x^2}=$ _________ .
View full solution →If function $f(x)=\frac{1-\cos (c x)}{x \sin x}, x \neq 0$ and $f(0)=\frac{1}{2}$ and $f(x)$ is continuous at $x=0$ then find value of $c$.
View full solution →Find the value of $\frac{d y}{d x}$ at $\theta=\frac{\pi}{4}$ when :
$x=a e^\theta(\sin \theta-\cos \theta)$
$y=a e^\theta(\sin \theta+\cos \theta)$
View full solution →Function $f(x)=\left\{\begin{array}{c}x^2+m, \text { when } x \neq 0 \\ -x^2-m, \text { when } x=2\end{array}\right.$, is continuous at $x=0$ then find value of $m$.
View full solution →If given function is continuous at $x=2$. Find the value of $k$ :$
f(x)=\left\{\begin{array}{cc}
\frac{x^3+x^2-16 x+20}{(x-2)^2}, & \text { when } x \neq 2 \\
k, & \text { when } x=2
\end{array}\right.
$
View full solution →$f(x)=\left\{\begin{array}{cl}\frac{x e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}, & x \neq 0 \\ 0, & x=0\end{array}, x=0\right.$ Examine the continuity at $x =0$.
View full solution →