Continuity and ...

Differentiate $$\log(1+x^{2})$$ with respect $$\tan^{-1}x$$.

An angle $$\theta$$ through which a pulley turns with time $$'t'$$ is completed by $$\theta = t^{2} + 3t - 5\ sq. cms/ min$$. Then the angular velocity for $$t = 5\ sec$$.

If $$f\left( x \right) = \left( {x - a} \right)g\left( x \right)$$ and $$g\left( x \right)$$ is continuous $$x = a$$ then $${f^{'}}\left( 1 \right) = $$

Let $$g\left(x\right)=\dfrac{f\left(x\right)}{x+1}$$ where $$f\left(x\right) $$ is differentiable on  $$\left[0,5\right]$$ such that $$f\left(0\right)=4,f\left(5\right)=-1$$. There exists $$c\in \left(0,5\right)$$ such that $$g^{ ' }\left( c \right) $$ is ?

Let $$f(x)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7$$.
The value of $$\lim _{ h\rightarrow 0 }{ \dfrac { f\left( 1-h \right) -f\left( 1 \right)  }{ { h }^{ 3 }+3h }  } $$ is

If $$y=y(x)$$ and it follows the relation $$e^{xy^{2}}+y\cos(x^{2})=5$$ then $$y'(0)$$ is equal to

A function $$f:R\rightarrow R$$ is such that $$f(1)=3$$ and $$f'(1)=6$$. Then $$\rightarrow { \displaystyle \lim _{ x\rightarrow 0 }{ \left[ \dfrac { f\left( 1+x \right)  }{ f\left( 1 \right)  }  \right]  }  }^{ 1/x }=$$ ?

If $$f$$ is a real valued differentiable function satisfying $$\left| f\left( x \right) -f\left( y \right)  \right| \le { \left( x-y \right)  }^{ 2 },x,y\epsilon R\ and f\left( 0 \right) ,\ then f\left( 1 \right)$$ equals ?

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