Algebra of Derivatives

Q.

The value of $\underset{x\to 1^{-}}{lim}\left[sinsi{n}^{-1}x\right]$ is

1. does not exist

2. π/2

3. 0

4. 1

Q.

If P(x) be a polynomial of degree 4, with P(2)=-1, P'(2)=0, P”(2)=2, P”'(2)=-12 and P^ir(2) =24, then P”(1) is equal to

1. 28

2. 24

3. 22

4. 26

Q.

For a function given by $y=f\left(t\right)\text{and}x=g\left(t\right)$. Then$\frac{dy}{dx}$ can be expressed as
$\frac{\frac{df\left(t\right)}{dt}}{\frac{dg\left(t\right)}{dt}}$

1. True

2. False

Q. 24. (ax2 + sinx)(p+ qcosx)

Q. Find the derivative of the following:
$x^5(3-6x^{-9})$.

Q. Find the derivative of for some fixed real number a .

Q.

Given f(x) = g(X) . h (x) and $f^{{}^{\prime }}\left(x\right)$=$g^{{}^{\prime }}\left(x\right)$h(x) + g(x)$h^{{}^{\prime }}\left(x\right)$ find f'(x) where f(x) = x sin x.

1. x sin x + cos x

2. sin x + x cos x

3. sin x + cos x

4. sin x - x cos x

Q. Find the derivative of the following functions from first principle
$\frac{1}{x^2}$

Q. 19. sin"x

Q. 22. x*(5sinx - 3cosx)

Q.

If tan $tan22{\frac{1}{2}}^{\circ }=x$ what is the value of tan${135}^{\circ }$?

1. 

2. 

3. 

4. 

Q. The slope of the normal to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (–10, –1) is

वक्र x = t² + 3t – 8, y = 2t² – 2t – 5 के बिंदु (–10, –1) पर अभिलम्ब की प्रवणता है

1. –6
2. $-\frac{1}{6}$
3. $\frac{1}{6}$
4. 6

Q.

If a, b, c are in A.P.; b, c, d are in G.P. and $\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P. prove that a, c, e are in G.P.

Q. For some constants a and b , find the derivative of (i) ( x – a ) ( x – b ) (ii) ( ax 2 + b ) 2 (iii)

Q.

For a function given by $y=f\left(t\right)\text{and}x=g\left(t\right)$. Then$\frac{dy}{dx}$ can be expressed as
$\frac{\frac{df\left(t\right)}{dt}}{\frac{dg\left(t\right)}{dt}}$

1. False

2. True

Q. Find the derivative of $\frac{x^n-a^n}{x-a}$ for some constant $a$

Q. Find the derivative of $99x$ at $x=100$

Q. sin x +coS x17sin x-cosx

Q. If $x=sint,y=tcost,\frac{dy}{dx}=$.

1. 1 + t tant
2. 1 - t tant
3. 1-tant
4. tant - t tant

Q.

If y=$co{s}^2{x}^2,find\frac{dy}{dx}.$

Q.

$\text{Let}f\left(x\right)=e^x,g=si{n}^{-1}xandh\left(x\right)=f\left(g\right(x\left)\right),then\frac{h^{\prime }\left(x\right)}{h\left(x\right)}$

1. 

2. 

3. 

4. 

Q. Find the derivative of the following functions (it is to be understood that a , b , c , d , p , q, r and s are fixed non-zero constants and m and n are integers):

Q. For the function Prove that

Q.

Given $f\left(x\right)=\frac{sinx}{x}$. Find f'(x).

1. 
2. 

3. 

4. 

Q.

If P(x) be a polynomial of degree 4, with P(2)=-1, P'(2)=0, P”(2)=2, P”'(2)=-12 and P^ir(2) =24, then P”(1) is equal to

1. 22

2. 24

3. 26

4. 28

Q.

If a function is of form
$f\left(z\right)=\frac{g\left(z\right)}{h\left(z\right)}$, then
$f^{\prime }\left(z\right)=\frac{\frac{\left(h\right(z\left)\right)d\left(g\right(z\left)\right)}{dz}+g\left(z\right)\frac{d\left(h\right(z\left)\right)}{dz}}{\left(h\right(z){)}^2}$

1. True
2. False

Q. Let f be a function such that f(x+y)=f(x)+f(y) for all x and y and $f\left(x\right)=(2x^2+3x)g\left(x\right)$ for all x where g(x) is continuous and g(0)=9 then f'(0) is equals to

1. 9
2. 6
3. 3
4. 27

Q. Given that $f\left(x\right)=\frac{sinx}{x}$. Then f'(x)=.

1. $\frac{cosx}{x}$
2. $-\frac{sinx}{x^2}$
3. $\frac{xcosx-sinx}{x^2}$
4. $\frac{xsinx+cosx}{x^2}$

Q. If $F\left(x\right)=f\left(x\right).g\left(x\right)and{f}^{\prime }\left(x\right)g^{\prime }\left(x\right)=c$ , where ‘c’ is a constant then $\frac{f‘‘}{f}+\frac{g‘‘}{g}+\frac{2C}{fg}$=

1. $\frac{f^{\prime \prime }}{F}$
2. $\frac{f^{\prime \prime }}{F^{\prime }}$
3. $\frac{F^{\prime \prime }}{F}$
4. $Noneofthese$

Q. For some constant $a$ and $b$, find the derivative of the following functions:
$(a{x}^2+b{)}^2$.