Sample QuestionsDifferentiation of Derivatives questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The derivative of $2 x^4+x$ is
- ✓
$8 x^3+1$
- B
$8 x^3$
- C
$8 x^3-1$
- D
$8 x^2+1$
Answer: A.
View full solution →If $f(x)=1+x+\frac{x^2}{2}+\ldots+\frac{x^{100}}{100}$, then $f^{\prime}(1)$ is equal to :
- A
$\frac{1}{100}$
- ✓
$1 0 0$
- C
$0$
- D
Answer: B.
View full solution →If $x=t^2$ and $y=t^3$ then $\frac{d^2 y}{d x^2}$ is
- A
$\frac{3}{2}$
- ✓
$\frac{3}{4 t}$
- C
$\frac{3}{2 t}$
- D
$\frac{3}{4}$
Answer: B.
View full solution →If $f(x)=\frac{x-4}{2 \sqrt{x}}$, then $f^{\prime}(1)$ is equal to :
- ✓
$\frac{5}{4}$
- B
$\frac{4}{5}$
- C
- D
$0$
Answer: A.
View full solution →If $y=\sqrt{x}+\frac{1}{\sqrt{x}}$, then $\frac{d y}{d x}$ at $x=1$ is equal to:
- A
- B
$\frac{1}{2}$
- C
$\frac{1}{\sqrt{2}}$
- ✓
$0$
Answer: D.
View full solution →If $x=a \sin p t, y=b \cos p t$, then find $\frac{d y}{d x}$ at $t=0$.
View full solution →Differentiate $3^x+x^3+4 x-5$ with respect to $x$
View full solution →Differentiate $\frac{x}{\sin x}$ with respect to $x$.
View full solution →Find the value of $k$ for which the function.
$
f(x)=\left\{\begin{array}{cc}
\frac{x^2+3 x-10}{x-2}, & x \neq 2 \\
k, & x=2
\end{array}\right.
$
is continuous at $x=2$.
View full solution →Show that $\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$ does not exist.
View full solution →Differentiate $f(x)=x^5 e^x+x^3 \log x-2^x$ with respect to $x$.
View full solution →Find $\lim _{x \rightarrow 1} f(x)$, where $f(x)=\left\{\begin{array}{cc}x^2-1, & x \leq 1 \\ -x^2-1, & x>1\end{array}\right.$
View full solution →If $\left(x^2+y^2\right)^2=x y$, find $\frac{d y}{d x}$.
View full solution →Differentiate $e^x \sin x+x^n \cos x$ with respect to $x$.
View full solution →$\lim _{x \rightarrow 3}\left(\frac{x^4-81}{2 x^2-5 x-3}\right)$
View full solution →Differentiate $x^{\sin x}+(\sin x)^{\cos x}$ with respect to $x$.
View full solution →Find the first principle derivative of $\frac{a x+b}{c x+d}$.
View full solution →Find the first principle the derivative of $\frac{x+1}{x-1}$.
View full solution →If
$f(x)=\left\{\begin{array}{cl}\frac{\sin (a+1) x+2 \sin x}{x}, & x<0 \\ 2, & x=0 \\ \frac{\sqrt{1+b x}-1}{x}, & x>0\end{array}\right.$
is continuous at $x=0$, then find the values of $a$ and $b$.
View full solution →Find the value of ' $a$ ' and ' $b^{\prime}$ if $\lim _{x \rightarrow 2} f(x)$ and $\lim _{x \rightarrow 4} f(x)$ exists where
$f(x)=\left\{\begin{array}{cc}x^2+a x+b, & 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b, & 4 < x \leq 8\end{array}\right.$
View full solution →The derivative of $f(x)=x \cos x$ is _________________
View full solution →If $y=\sqrt{\sin x+y}$, then $\frac{d y}{d x}$ is equal to _________________
View full solution →If $y=\frac{1+\frac{1}{x^2}}{1-\frac{1}{x^2}}$, then $\frac{d y}{d x}$ is equal to _________________
View full solution →If $y=\frac{1}{x^{11}}$ then $\frac{d y}{d x}=$_________________
View full solution →If $y=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!} \ldots$, then $\frac{d y}{d x}=$_________________
View full solution →Find the first principle the derivative of $x^3-27$.
View full solution →If $x=a(\cos 2 \theta+2 \theta \sin 2 \theta)$ and $y=a(\sin 2 \theta-2 \theta$ $\cos 2 \theta)$, find $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{8}$
View full solution →If $y=a e^{2 x}+b e^{-x}$, then show that $\frac{d^2 y}{d x^2}-\frac{d y}{d x}-2 y=0$.
View full solution →If $y=\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}$, prove that $(2 x y) \frac{d y}{d x}=\frac{x}{a}-\frac{a}{x}$.
View full solution →If $x \sqrt{1+y}+y \sqrt{1+x}=0$ and $x \neq y$ prove that $\frac{d y}{d x}=-\frac{1}{(x+1)^2}$.
View full solution →| (a) $\frac{d}{d x}(1)$ | (i) $\frac{1}{x}$ |
| (b) $\frac{d}{d x}\left(x^n\right)$ | (ii) 1 |
| (c) $\frac{d}{d x}(x)$ | (iii) 0 |
| (d) $\frac{d}{d x}\left(\log _e x\right)$ | (iv) $n x^{n-1}$ |
| (a) $\lim _{x \rightarrow 0} \frac{\sin (x-a)}{(x-a)}$ | (i) $e$ |
| (b) $\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}$ | (ii) 1 |
| (c) $\lim _{x \rightarrow c} e^x$ | (iii) 0 |
| (d) $\lim _{x \rightarrow 0} \tan x$ | (iv) $e^c$ |