\(\frac{d}{d x}\left(\frac{2}{\pi} \sin x^{\circ}\right)\) is

(a)

\({\pi\over 180}cos \ x^o\)

(b)

\({1\over 90} cos \ x^o\)

(c)

\({\pi\over 90}cos \ x^o\)

(d)

\({2\over \pi}cos \ x^o\)

If y = f(x²+2) and f '(3) = 5, then \({dy\over dx}\) at x = 1 is

(a)

5

(b)

25

(c)

15

(d)

10

If x = a sin \(\theta\) and y = b cos \(\theta\), then \({d^2y\over dx^2}\)is

(a)

\({a \over b^2}sec^2 \theta\)

(b)

\(-{b \over a}sec^2 \theta\)

(c)

\(-{b \over a^2}sec^3 \theta\)

(d)

\(-{b^2\over a^2}sec^3 \theta\)

The differential coefficient of log₁₀ x with respect to log_x10 is

(a)

1

(b)

-(log₁₀ x)²

(c)

(log_x 10)²

(d)

\(x^2\over100\)

If f(x) = x + 2, then f '(f(x)) at x = 4 is

(a)

8

(b)

1

(c)

4

(d)

5

\(\text { If } f(x)=\left\{\begin{array}{ll} a x^2-b, & -1<x<1 \\ \frac{1}{|x|}, & \text { elsewhere } \end{array} \ \text { is differentiable at } x=1\right. \text {, then }\) 

(a)

\(a={1\over2},b={-3\over 2}\)

(b)

\(a={-1\over2},b={3\over 2}\)

(c)

\(a=-{1\over2},b=-{3\over 2}\)

(d)

\(a={1\over2},b={3\over 2}\)

The number of points in R in which the function \(f(x)=|x-1|+|x-3|+sin \ x\) is not differentiable, is

(a)

3

(b)

2

(c)

1

(d)

4

\(\int \sin ^3 x d x\) is

(a)

\({-3\over 4}cos \ x-{cos \ 3x\over 12}+c\)

(b)

\({3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(c)

\({-3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(d)

\({-3\over 4}sin \ x-{sin \ 3x\over 12}+c\)

\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\) is

(a)

x+c

(b)

\({x^3\over 3}+c\)

(c)

\({3\over x^3}+c\)

(d)

\({1\over x^2}+c\)

\(\int \frac{\sec x}{\sqrt{\cos 2 x}} d x\) is

(a)

tan^-1 (sin x)+c

(b)

2sin^-1(tan x)+c

(c)

tan^-1(cos x)+c

(d)

sin ^-1(tan x)+c

Find the derivatives of the following functions using first principle. f(x) = - x² + 2

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
\(f(x)=|x-1|\)

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
\(f(x)=\sqrt{1-x^2}\)

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

Differentiate the following with respect to x : y = x³ + 5x² + 3x + 7

Differentiate the following with respect to x : y = e^x + sin x + 2

Differentiate the following with respect to x : y = 4 cosec x - log x - 2e^x

Differentiate the following with respect to x : \(y=(x-{1\over x})^2\)

Differentiate the following: y = (x² + 4x + 6)⁵

Differentiate the following: y = tan 3x

Differentiate the following: \(y=\sqrt{1+2 \ tan \ x}\)

Differentiate the following: y = sin³ x + cos³ x

Integrate the following with respect to x : e^x

Integrate the following with respect to x : \((1+x^2)^{-1}\)

Integrate the following with respect to x : \(\left(1-x^2\right)^{-\frac{1}{2}}\)