Derivative Definition

Q.

What is the formula of $\left(a^3+b^3\right)$?

Q. Show that the function defined by $g\left(x\right)=x-\left[x\right]$ is discontinuous at all integral points. Here $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Q. Let $f:R\to R$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right)+x^{2}y+x{y}^2$ $\forall x,y\in R$. If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=1$, then $f\left(x\right)$ is

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

Q. Let $f(x+y)=f\left(x\right)\cdot f\left(y\right)\forall x,y\in R.$ If $f\left(5\right)=2$ and $f^{\prime }\left(0\right)=3,$ then the value of $f^{\prime }\left(5\right)$ is

Q. $f\left(x\right)=x^2$ in $2\le x\le 3$ Is Rolle's theorem applicable?

Q. If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k{)}^6$ is equal to

Q. If $y=(1+x)(1+2x)(1+3x),$ then the value of $\frac{dy}{dx}$ at $x=0$ is

[1 mark]

1. $5$
2. $6$
3. $2$
4. $0$

Q. If $3^{x/2}+2^x>25$ then the solution set is

1. $R$
2. $(2,+\infty )$
3. $(4,+\infty )$
4. None of these

Q. The solution of $(1+y+x^{2}y)dx+(x+x^3)dy=0$ is:

1. $y+{\tan }^{-1}x=C$
2. $xy+{\tan }^{-1}x=C$
3. $x^2+{\tan }^{-1}y/x=C$
4. $y^2+{\tan }^{-1}x=C$

Q.

If the length of the tangent drawn at the point $(1,3)$ on the curve $y=3x^3$ is $a$, then find the value of $9a^2$

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Q. Let $f(x+y)=f\left(x\right)\cdot f\left(y\right)\forall x,y\in R.$ If $f\left(5\right)=2$ and $f^{\prime }\left(0\right)=3,$ then the value of $f^{\prime }\left(5\right)$ is

Q. If $y=(1+x)(1+2x)(1+3x),$ then the value of $\frac{dy}{dx}$ at $x=0$ is

[1 mark]

1. $0$
2. $2$
3. $6$
4. $5$

Q. Let $f:R\to R$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right)+x^{2}y+x{y}^2$ $\forall x,y\in R$. If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=1$, then $f\left(x\right)$ is

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

Q. Let $f:R\to R$ be a function such that $f(x+y)=f\left(x\right)+f\left(y\right)+x^{2}y+x{y}^2$ $\forall x,y\in R$. If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=1$, then $f\left(x\right)$ is

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

Q. Let $f(x+y)=f\left(x\right)\cdot f\left(y\right)\forall x,y\in R.$ If $f\left(5\right)=2$ and $f^{\prime }\left(0\right)=3,$ then the value of $f^{\prime }\left(5\right)$ is

Q. Differentiate from first principle:

$\left(iii\right)\frac{1}{x^3}$