Theorems for Differentiability

Q.

The derivative of sin y is:

1. 

2. 

3. 

4. 

Q.

The period of y =sin 5x is

1. 

2. 

3. 

4. 

Q. Let $f(x+y)=f\left(x\right)+f\left(y\right)$ and $f\left(x\right)=x^{2}g\left(x\right)$ for all $x,yϵR$, where $g\left(x\right)$ is continuous function. Then $f^{\prime }\left(x\right)$ is equal to

1. g'(x)
2. g(0)
3. g(0) + g'(x)
4. 0

Q. Y = f( 1 /x) and f(x) = sin( x ^2) then dy/dx is

Q.

The following expansion can be used to find the difficult limits easily. We recommend you to remember them.

$1)e^x=\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+......$

$2\left)ln\right(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}.....,$ -1 < x $\ge$ 1

$3)sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}.....$

$4)cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......$

$5)tanx=x+\frac{x^3}{3!}+\frac{2x^5}{15!}......$

$6)a^x=1+(lna)x+(lna{)}^2\frac{x^2}{2!}.......$

We will be using these expansions to solve problems in this topic.

If you are done with reading this, click True to go to questions.

1. True

2. False

Q. 3. Find the common roots of the equations }2\operatorname{sin}^2x+\operatorname{sin}^22x=2 and }\operatorname{sin}2x+\operatorname{cos}2x=\operatorname{tan}x

Q. If f(x)={|x+2|/tan^-1(x+2), x≠-2;2, x=-2 then discuss the continuity of f(x) at x=-2

Q. Solve }\vert-2x^2+1+e^x+\operatorname{sin}x\vert=\vert2x^2-1\vert+e^x+\vert\operatorname{sin}x\vert, x∈\lbrack0, 2π\rbrack

Q. If sin(x+y)dy/dx=5 then

Q. Solve the equation }3^{\operatorname{sin}2x+2\operatorname{cos}^2x}+3^{1-\operatorname{sin}2x+2\operatorname{sin}^2x}=28

Q. calculas differentiation: y=sin^3(4x^2+2) find dy/dx=?

Q. Evaluate the following one sided limits:

(i)
$\underset{x\to 2^{+}}{\lim }\frac{x-3}{x^2-4}$

(ii)
$\underset{x\to 2^{-}}{\lim }\frac{x-3}{x^2-4}$

(iii)
$\underset{x\to 0^{+}}{\lim }\frac{1}{3x}$

(iv)
$\underset{x\to -8^{+}}{\lim }\frac{2x}{x+8}$

(v)
$\underset{x\to 0^{+}}{\lim }\frac{2}{x^{1/5}}$

(vi)
$\underset{x\to \frac{\pi }{2}}{\lim }\tan x$

(vii)
$\underset{x\to -\pi /2^{+}}{\lim }\sec x$

(viii)
$\underset{x\to 0^{-}}{\lim }\frac{x^2-3x+2}{x^3-2x^2}$

(ix)
$\underset{x\to -2^{+}}{\lim }\frac{x^2-1}{2x+4}$

(x)
$\underset{x\to 0^{-}}{\lim }2-\cot x$

(xi)
$\underset{x\to 0^{-}}{\lim }1+\mathrm{cosec}x$

Q. Let f(x + y) = f(x)f(y) and f(x) = 1 + sin(3x)g(x) where g(x) is continuous then f'(x) is
[Kerala (Engg.) 2005]

1. f(x)g(0)
2. 3g(0)
3. f(x)cos 3x
4. 3 f(x)g(x)
5. 3 f(x) g(0)

Q.

If $A=\{x:x=3n,nϵZ\}$ and $B=\{x:x=4n,nϵZ\}$ then find $A\cap B$.

Q. Let $a,bϵR$ and $f:R\to R$ be defined by $f\left(x\right)=acos(|x^3-x|)+b|x|sin(|x^3+x|)$.
Then $f$ is

1. Differentiable at x=0 if a =0 and b=1
2. Differentiable at x=1 if a =1 and b=0
3. Differentiable at x=0 if a =1 b=0
4. Differentiable at x=1 if a =1 b=1

Q.

The general solution of sin = 0 is

1. nπ where n is a real number

2. 2π

3. π

4. nπ, where n is an integer

Q. If y=X sinx² then the value of Dy/DX is

Q.

$\underset{x\to \frac{\pi }{2}}{lim}\frac{sinx-(sinx{)}^{sinx}}{1-sinx+1nsinx}$ is equal to

1. 4

2. 2

3. -1

4. 0

Q.

(i) If $f\left(x\right)=\{\begin{array}{cc}\frac{x-\left|x\right|}{x}& ifx\ne 0\\ 2& ifx=0\end{array}$, show that $\underset{x\to 0}{lim}f\left(x\right)$ does not exist.

(ii) Evaluate $\underset{x\to 0}{lim}\frac{sinx-2xin3x+sin5x}{x}.$

Or

(i) Find the derivative of $\frac{(x-1)(x-2)}{(x-3)(x-4)}.$

(ii) Differentiate $x{e}^x$ by using first principle.

Q. If f(x) = $\{\begin{array}{cc}\sin x& x\ne n\pi ,n=0,\pm 1,\pm \mathrm{2...}\\ 2,& otherwise\end{array}$ and g(x) = $\{\begin{array}{cc}{x}^2+1,& x\ne 0,2\\ 4,& x=0\\ 5,& x=2\end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$ is

1. 
2. 
3. 
4. 

Q. $\underset{x\to 1}{\lim }{\left\{\frac{x^3+2x^2+x+1}{x^2+2x+3}\right\}}^{\frac{1-\cos \left(x-1\right)}{{\left(x-1\right)}^2}}$

Q. $\underset{x\to 0}{\lim }\frac{3\sin x-\sin 3x}{x^3}$

Q. If f(x) = sin [π²] x + sin [−π]² x, where [x] denotes the greatest integer less than or equal to x, then

(a) f(π/2) = 1
(b) f(π) = 2
(c) f(π/4) = −1
(d) None of these

Q. The value of $2\tan \frac{\pi }{10}+3\sec \frac{\pi }{10}-4\cos \frac{\pi }{10}$ is
(a) 0
(b) $\sqrt{5}$
(c) 1
(d) none of these

Q. calculas differentiation: y=sin^3(4x^2+2) find dy/dx=?

Q. Let $a,bϵR$ and $f:R\to R$ be defined by $f\left(x\right)=acos(|x^3-x|)+b|x|sin(|x^3+x|)$.
Then $f$ is

1. Differentiable at x=0 if a =0 and b=1
2. Differentiable at x=1 if a =1 and b=0
3. Differentiable at x=0 if a =1 b=0
4. Differentiable at x=1 if a =1 b=1

Q. Find the derivative of the following functions from first principle:
$\cos (x-\frac{\pi }{8})$

Q. $\underset{x\to \mathrm{\pi }/2}{\lim }\frac{2^{-\cos x}-1}{x\left(x-{\displaystyle \frac{\mathrm{\pi }}{2}}\right)}$

Q.

find the derivative of xcosx

Q.

Let $a_1,a_2,....a_n$be fixed real numbers and let

$f\left(x\right)=(x-a_1)(x-a_2)(x-a_3)...(x-a_n)$.

Find $\underset{x\to a_1}{lim}f\left(x\right),$

If $a\ne a_1,a_2,...a_n,compute\underset{x\to a}{lim}f\left(x\right)$