Algebra of Derivatives

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.

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

1. 
2. 
3. 
4. 

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 $f\left(x\right)=\tan x$. Then $f^{\prime }\left(x\right)$ is equal to

1. ${\sec }^{2}x$

2. ${\text{cosec}}^{2}x$

3. $-{\sec }^{2}x$

4. $x{\sec }^{2}x$

Q.

If $x=\sin t,y=t\cos t.$ Then $\frac{dy}{dx}$ is equal to

1. $1+t\tan t$

2. $1-t\tan t$

3. $1-\tan t$

4. $\tan t-t\tan t$

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. 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 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