If \(y={1\over a-z}\) , then \({dz\over dy}\) is

(a)

\((a-z)^2\)

(b)

-(z - a)²

(c)

(z + a)²

(d)

-(z + a)²

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

(a)

1

(b)

2

(c)

3

(d)

-3

\({d\over dx}(e^{x+5log \ x})\) is

(a)

e^x.x⁴(x+5)

(b)

e^x.x(x+5)

(c)

e^x\(+{5\over x}\)

(d)

e^x\(-{5\over x}\)

\(x={1-t^2\over 1+t^2},y={2t\over 1+t^2}\) then \({dy\over dx}\)is

(a)

\(-{y\over x}\)

(b)

\({y\over x}\)

(c)

\(-{x\over y}\)

(d)

\({x\over y}\)

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 \(y={(1-x)^2\over x^2}\), then \({dy \over dx}\) is

(a)

\(\frac{2}{x^2}+\frac{2}{x^3}\)

(b)

\(-\frac{2}{x^2}+\frac{2}{x^3}\)

(c)

\(-\frac{2}{x^2}-\frac{2}{x^3}\)

(d)

\(-\frac{2}{x^3}+\frac{2}{x^2}\)

If pv = 81, then \({dp\over dv}\) at v = 9 is

(a)

1

(b)

-1

(c)

2

(d)

-2

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

The derivative of f(x) = x |x| at x = −3 is

(a)

6

(b)

-6

(c)

does not exist

(d)

0

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

(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)

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

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) = - x² + 2

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

Find the derivatives of the following functions with respect to corresponding independent variables: g(t) = t³cos t

Differentiate the following: F(x) = (x³ + 4x)⁷

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

Determine whether the following function is differentiable at the indicated values. f(x) = sin|x| at x = 0

Find the derivatives of the following functions with respect to corresponding independent variables: \(y=\frac{\tan x-1}{\sec X}\)

Find F'(x) if F(x) = \(\sqrt{x^2+1}\)

Differentiate the following: y = e^−mx

Find the slope of tangent line to the graph of f(x) = - 5x² + 7x at (5, f(5)).

Find the derivatives of the following functions with respect to corresponding independent variables: y = e^-x. log x

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