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  • Confused! kindly explain, - Letbe a differentiable function such that andfor allIfthen is equal to : - Limit , continuity and differentiability - JEE Main

Let f be a differentiable function such that f\left ( 1 \right )=2  and  {f}'\left ( x \right )=f(x) for all x\equiv \mathbf{R}. If h(x)=f(f(x)), then {h}'\left ( 1 \right ) is equal to : 

  • Option 1)

    4e^{2}

     

  • Option 2)

    2e

  • Option 3)

    4e

  • Option 4)

    2e^{2}

Answers (1)

best_answer

 

Derivative at a point -

The value of  f'(x) obtained by putting  x = a is called the derivative of  f(x) at  x = a  and it is denoted by  f'(a)  or  

\frac{dy}{dx}     at  x = a.

-

\frac{dt}{dx}=f\\\\log\left | f \right |=x+c\\\\\\\\x=1\:\; \; \; t-=2\\\\\\log\: 2=1+c\; \; \; c=log\: 2-1\\\\log\left | f \right |=x+log\: 2-1\\\\\Rightarrow \frac{\left | f \right |}{2}=e^{x-1}\Rightarrow f=2\: e^{x-1}\\\\as\: \: \: \: f=0 \\\\

 

 

{h}'(x)=f^{'}(f(x))\times f^{'}(x)\\\\\: {h}'(1)={f}'(2)\times {f}'(1)\\\\\: {f}'(1)=f(1)=2\\\\\: {f}'(2)=2e^{2-1}=2e\\\\\: \Rightarrow {h}'(1)=2e\: \times 2=4e

 


Option 1)

4e^{2}

 

Option 2)

2e

Option 3)

4e

Option 4)

2e^{2}

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