Basic Differentiation Rule

Q.

The inverse of $y=5^{\log \left(x\right)}$ is

1. $x=5^{\log \left(y\right)}$

2. $x=5^{\log \left(y\right)}c$

3. $x=y^{\frac{1}{\log \left(5\right)}}$

4. $x=5^{\frac{1}{\log \left(y\right)}}$

Q.

The inverse of the function $y=\frac{{10}^x-{10}^{-x}}{{10}^x+{10}^{-x}}$ is

1. $\left(\frac{1}{2}\right){\log }_{10}\left(\frac{x+1}{x-1}\right)$

2. $\left(\frac{1}{2}\right){\log }_{10}\left(\frac{2+x}{2-x}\right)$

3. $\left(\frac{1}{2}\right){\log }_{10}\left(\frac{1-x}{1+x}\right)$

4. None of these

Q. If $S=ut+\frac{1}{2}a{t}^2$, where $S$ is displacement, $u$ is initial velocity (constant), $a$ is acceleration (constant) and $t$ is time taken, then differentiation of $S$ w.r.t. $t$ will be

1. $u+\frac{at}{2}$
2. $u+2at$
3. $\frac{u{t}^2}{2}+\frac{a{t}^3}{6}$
4. $u+at$

Q. Where does the curve $y=x^4-2x^2+2$ have any horizontal tangents?

1. $(0,2),(1,1)$ and $(-1,1)$
2. $(0,0),(-1,1)$ and $(-1,3)$
3. $(3,4),(2,1)$ and $(-1,4)$
4. $(0,4),(3,1)$ and $(-2,4)$

Q. If $y=1/x^4,$ then $\frac{dy}{dx}$ will be

1. $\frac{4}{x^3}$
2. $4x^3$
3. $-\frac{4}{x^5}$
4. $\frac{4}{x^5}$

Q. Derivative of $f\left(x\right)=4\sqrt{x}+\frac{1}{\sqrt{x}}$ is

1. $\frac{2}{\sqrt{x}}+\frac{1}{2x\sqrt{x}}$
2. $\frac{2}{\sqrt{x}}-\frac{1}{2x\sqrt{x}}$
3. $\frac{1}{2x\sqrt{x}}-\frac{2}{\sqrt{x}}$
4. $-\frac{1}{2x\sqrt{x}}-\frac{2}{\sqrt{x}}$

Q.

The minimum value of $f\left(x\right)=2x^3-21x^2+36x-20$ is

1. $-128$

2. $-126$

3. $-120$

4. None of these

Q.

Let $f:R\to R$ be a function defined by $f\left(x\right)=\frac{\left[e^x-e^{-x}\right]}{\left[e^x+e^{-x}\right]}$ then, $\left(1\right)$ is both one-one and onto $\left(2\right)f$ is one-one but onto

1. $f$ is both one-one and onto

2. $f$ is one-one but not onto

3. $f$ is onto but not one-one

4. $f$ is neither one-one nor onto

Q. Derivative of $f\left(x\right)=\frac{x^2}{\cos x}$ is

1. $\frac{2x\cos x+x^2\sin x}{\sin x}$
2. $\frac{2x\cos x-x^2\sin x}{{\cos }^{2}x}$
3. $\frac{2x\cos x+x^2\sin x}{\cos x}$
4. $\frac{2x\cos x+x^2\sin x}{{\cos }^{2}x}$

Q. Differentiation of $\frac{-x^3}{3}$ w.r.t $x$ is

1. $x^2$
2. $-x^2$
3. $-3x^2$
4. $3x^2$

Q. If $y=x^2\sin x$, then $\frac{dy}{dx}$ will be

1. $x^2\cos x+2x\sin x$
2. $2x\sin x$
3. $x^2\cos x$
4. $2x\cos x$

Q. Find the slope of the tangent to the curve $y=2\sqrt{x}$ at the point $(2,1)$.

1. $\frac{1}{\sqrt{5}}$
2. $\frac{1}{\sqrt{2}}$
3. $\sqrt{3}$
4. $\sqrt{5}$

Q. If $f\left(x\right)=\sqrt{x}g\left(x\right)$, where $g\left(4\right)=2$ and $g^{\prime }\left(4\right)=3$, find $f^{\prime }\left(4\right)$.

1. 5
2. 6.5
3. 16
4. 4

Q. If $y=x^4+5e^x+9$, then $\frac{d^{2}y}{d{x}^2}$ is

1. $4x^3+5e^x$
2. $\frac{x^3}{4}+\frac{e^x}{5}$
3. $4x^3+5e^x+9$
4. $12x^2+5e^x$

Q. Find the slope of the tangent to the curve $y=4x^2-3x-8$ at point $(-3,4)$.

1. $27$
2. $-27$
3. $-29$
4. $29$

Q. Find the maximum and minimum value of function $f\left(x\right)=x^3-6x^2+9x+1$ in the interval $x=0\text{to}x=5$.

1. 21 and 1
2. 20 and 2
3. 16 and 14
4. 18 and 2

Q. Differentiate $y=\text{cosec}x\cot x$ w.r.t $x$

1. $\text{cosec}x({\text{cosec}}^{2}x+{\cot }^{2}x)$
2. $\text{cosec}x$
3. $-\text{cosec}x({\text{cosec}}^{2}x+{\cot }^{2}x)$
4. $\text{cosec}x({\text{cosec}}^{2}x-{\cot }^{2}x)$

Q. If $y=2\sin x$, then $\frac{dy}{dx}$ will be

1. $2\cos x$
2. $-2\cos x$
3. $\cos x$
4. None

Q. Find the slope of the tangent to the curve $y=4x^2-3x-8$ at point $(-3,4)$.

1. $27$
2. $-27$
3. $-29$
4. $29$

Q.

If $f\left(x\right)\mathrm{and}g\left(x\right)$ are two functions with $g\left(x\right)=x-\frac{1}{x}$ and $fog\left(x\right)=x^3-\frac{1}{x^3}$, then $f\left(x\right)$ is equal to:

1. $3x^2+3$

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

3. $1+\frac{1}{x^2}$

4. $3x^2+\frac{3}{x^4}$

Q.

Find the value of $x$ if ${\left(\frac{5}{7}\right)}^6{\left(\frac{7}{5}\right)}^{-9}={\left(\frac{5}{7}\right)}^{3x}$

Q. Derivative of $f\left(x\right)=x^2\sin x$ is

1. $2x\sin x+x^2\cos x$
2. $2x\sin x+x\cos x$
3. $x\sin x+x^2\cos x$
4. $2x\sin x-x^2\cos x$

Q. Differentiate $y=\text{cosec}x\cot x$ w.r.t $x$

1. $\text{cosec}x({\text{cosec}}^{2}x+{\cot }^{2}x)$
2. $\text{cosec}x$
3. $-\text{cosec}x({\text{cosec}}^{2}x+{\cot }^{2}x)$
4. $\text{cosec}x({\text{cosec}}^{2}x-{\cot }^{2}x)$

Q. Slope of the curve $y=x^3-e^x+\cos x$ is

1. $x^2+e^x-\sin x$
2. $3x^2+e^x-\sin x$
3. $3x^2-e^x-\sin x$
4. $x^2-e^x+\sin x$

Q.

On the interval $\left[0,1\right]$, the function $x^{25}(1-x^{75})$ takes its maximum value at the point

1. $0$

2. $\frac{1}{4}$

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

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

Q.

$Ify=\frac{(x^2+2x)}{(3x-4)},then\frac{dy}{dx}=?$

1. $\frac{-3x^2+4x+8}{(3x-4{)}^2}$

2. $\frac{3x^2-8x-8}{(3x-4{)}^2}$

3. $\frac{3x^2+8x+8}{(3x-4{)}^2}$

4. none of these

Q. If surface area of a cube is changing at a rate of $18m^2/s$ , find the rate at which length of the body diagonal changes, at a moment when side length is $3m$.

1. $18\text{m/s}$
2. $3\sqrt{3}\text{m/s}$
3. $\frac{3}{2}\text{m/s}$
4. $\frac{\sqrt{3}}{2}\text{m/s}$

Q. If $S=e^x-\sin x$, find $\frac{d^{2}S}{d{x}^2}$

1. $e^x+\cos x$
2. $e^x-\sin x$
3. $e^x+\sin x$
4. $e^x-\cos x$

Q. If $y=x^2\sin \left(x\right)$, then $\frac{dy}{dx}$ will be

1. $x^2\cos \left(x\right)+2x\sin \left(x\right)$
2. $2x\sin \left(x\right)$
3. $x^2\cos \left(x\right)$
4. $2x\cos \left(x\right)$

Q.

Which of the following functions is inverse of itself?

1. $f\left(x\right)=\frac{1-x}{1+x}$

2. $f\left(x\right)=3logx$

3. $f\left(x\right)=3x(x+1)$

4. None of these