Addition Rule of Differentiation

Q. Let $f\left(x\right)=lo{g}_e\left\{\frac{u\left(x\right)}{v\left(x\right)}\right\},u^{\prime }\left(2\right)=4,v^{\prime }\left(2\right)=2,u\left(2\right)=2,v\left(2\right)=1$, then $f^{\prime }\left(2\right)$ is equal to

1. 0
2. 1
3. - 1
4. None of these

Q. If 5f(x) + 3f $\left(\frac{1}{x}\right)$ = x + 2 and y = x f(x), then ${\left(\frac{dy}{dx}\right)}_{x=1}$ is equal to

1. $14$
2. $\frac{7}{8}$
3. $1$
4. None of these

Q. If f(x) = $si{n}^{-1}\left(sinx\right)+co{s}^{-1}\left(sinx\right)and\varphi \left(x\right)=f\left(f\right(f\left(x\right)\left)\right),then{\varphi }^{\prime }\left(x\right)$ is equal to

1. 1
2. sin x
3. 0
4. None of these

Q.

If $y=ta{n}^{-1}\sqrt{\frac{1-sinx}{1+sinx}}$ then the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{6}$ is

1. $-\frac{1}{2}$

2. 1

3. $\frac{1}{2}$

4. -1

Q. Derivative of the function, $f\left(x\right)=2\ln {\left(x\right)}^5+2\ln {\left(x\right)}^3+\ln {\left(x\right)}^2-\ln \left(x\right)+2x$ is

1. $2$
2. $\frac{17}{x}+2$
3. $\frac{19}{x}+2$

Q. Derivative of the function, $f\left(x\right)=2\ln {\left(x\right)}^5+2\ln {\left(x\right)}^3+\ln {\left(x\right)}^2-\ln \left(x\right)+2x$ is

1. $2$
2. $\frac{17}{x}+2$
3. $\frac{19}{x}+2$

Q. Let $f\left(x\right)=lo{g}_e\left\{\frac{u\left(x\right)}{v\left(x\right)}\right\},u^{\prime }\left(2\right)=4,v^{\prime }\left(2\right)=2,u\left(2\right)=2,v\left(2\right)=1$, then $f^{\prime }\left(2\right)$ is equal to

1. 0
2. None of these
3. 1
4. - 1

Q. If f(x) = $si{n}^{-1}\left(sinx\right)+co{s}^{-1}\left(sinx\right)and\varphi \left(x\right)=f\left(f\right(f\left(x\right)\left)\right),then{\varphi }^{\prime }\left(x\right)$ is equal to

1. 1
2. sin x
3. None of these
4. 0

Q. If 5f(x) + 3f $\left(\frac{1}{x}\right)$ = x + 2 and y = x f(x), then ${\left(\frac{dy}{dx}\right)}_{x=1}$ is equal to

1. $14$
2. $\frac{7}{8}$
3. $1$
4. None of these

Q.

If $y=ta{n}^{-1}\sqrt{\frac{1-sinx}{1+sinx}}$ then the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{6}$ is

1. $-\frac{1}{2}$

2. $\frac{1}{2}$

3. 1

4. -1