If y = \({1\over4}u^4,u={2\over 3}x^3+5,\) then \({dy\over dx}\) is

(a)

\({1\over 27}x^2(2 x^3+15)^3\)

(b)

\({2\over 27}x(2 x^3+5)^3\)

(c)

\({2\over 27}x^2(2 x^3+15)^3\)

(d)

\(-{2\over 27}x(2 x^3+5)^3\)

If y = cos (sin x²), then \({dy\over dx}\) at x = \(\sqrt{\pi\over 2}\) is

(a)

-2

(b)

2

(c)

\(-2\sqrt{\pi\over 2}\)

(d)

0

If y = mx + c and f(0) =\(f '(0)=1\), then f(2) is

(a)

1

(b)

2

(c)

3

(d)

-3

If f(x) = x tan^-1 x, then f '(1) is

(a)

\(1+{\pi\over 4}\)

(b)

\({1\over 2}+{\pi\over 4}\)

(c)

\({1\over 2}-{\pi\over 4}\)

(d)

2

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

(a)

8

(b)

1

(c)

4

(d)

5

\(\text { If } f(x)= \begin{cases}x-5 & \text { if } x \leq 1 \\ 4 x^2-9 & \text { if } 1<x<2 \\ 3 x+4 & \text { if } x \geq 2\end{cases}\), then the right hand derivative of f(x) at x = 2 is

(a)

0

(b)

2

(c)

3

(d)

4

If g(x) = (x² + 2x + 3) f(x) and f(0) = 5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\), then g'(0) is

(a)

20

(b)

14

(c)

18

(d)

12

If f(x) = \(\left\{\begin{matrix} x+2& -1<x<3\\ 5,& x=3\\ 8-x,& x>3\\ \end{matrix}\right.\), then at x = 3, f'(x) is:

(a)

1

(b)

-1

(c)

0

(d)

does not exist

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

(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

Show that the greatest integer function \(f(x)=\left\lfloor x \right\rfloor \) is not differentiable at any integer?

Find the derivatives of the following functions using first principle. f(x) = 6

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

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

Differentiate the following with respect to x : \(y={log x \ x \over e^x}\)

Differentiate 2^x.

Differentiate the following: y = cos (tan x)

Find \({dy\over dx}\) if x = at² ; y = 2at, t\(\neq 0.\)

Find the derivatives of the following : y = x^cosx

Find the derivatives of the following : \(\sqrt{x^2+y^2}=tan^{-1}({y\over x})\)

Determine whether the following function is differentiable at the indicated values. f(x) = |x| + |x - 1| at x = 0, 1

Show that the following functions are not differentiable at the indicated value of x.

Find the derivatives of the following functions with respect to corresponding independent variables: \(y=\frac{x}{\sin X+\cos X}\)

Differentiate y \(=x^{\sqrt{x}}\)

Find f'(x) if f(x) = cos^-1(4x³ - 3x).

Find the slope of the tangent line to the graph of f(x) = 7x + 5 at any point (x₀, f(x₀)).

Examine the differentiability of functions in R by drawing the diagrams |sin x|