Question

If a polynomial in x is given by $f\left( x \right)$ is a polynomial in x, then the second derivative of $f\left( {{e}^{x}} \right)$ at $x=1$ is
(A). $ef''\left( e \right)+f'\left( e \right)$
(B). $\left( f''\left( e \right)+f'\left( e \right) \right){{e}^{2}}$
(C). ${{e}^{2}}f''\left( e \right)$
(D). $\left( f''\left( e \right)e+f'\left( e \right) \right)e$

Answer 1

Hint: Here, we should know the three important formula of derivative which is $\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}$ , $\dfrac{d}{dx}\left[ f\left( g\left( x \right) \right) \right]=g'\left( x \right)\cdot f'\left( g\left( x \right) \right)$ ; $\dfrac{d}{dx}\left[ g\left( x \right)\cdot h\left( x \right) \right]=g\left( x \right)\cdot h'\left( x \right)+h\left( x \right)\cdot g'\left( x \right)$ . We have to find second order derivative of $\dfrac{d}{dx}\left[ \dfrac{d}{dx}f\left( {{e}^{x}} \right) \right]$ . After getting the final answer we will replace the value of x as 1 wherever x is present in the answer.

Complete step-by-step solution -
Now, here we have to find second order derivative of $\dfrac{d}{dx}\left[ \dfrac{d}{dx}f\left( {{e}^{x}} \right) \right]$ . So, there are basically 3 formulas which will be used here.
$\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}$ ……………………………….(1)
$\dfrac{d}{dx}\left[ f\left( g\left( x \right) \right) \right]=g'\left( x \right)\cdot f'\left( g\left( x \right) \right)$ ………………………..(2)
$\dfrac{d}{dx}\left[ g\left( x \right)\cdot h\left( x \right) \right]=g\left( x \right)\cdot h'\left( x \right)+h\left( x \right)\cdot g'\left( x \right)$ ……………………………..(3)
So, first we will find a first order derivative i.e. $\dfrac{d}{dx}f\left( {{e}^{x}} \right)$ . So, we will use here equation(1) and (2) where $g\left( x \right)={{e}^{x}}$ . So, applying the formula we will get
$\dfrac{d}{dx}f\left( {{e}^{x}} \right)={{e}^{x}}\cdot f'\left( {{e}^{x}} \right)$ ………………………….(4)
Now, keeping this value of equation (4) into our main equation to find second derivative i.e.
$\dfrac{d}{dx}\left[ {{e}^{x}}\cdot f'\left( {{e}^{x}} \right) \right]$
So, here using equation (3) as we have 3 terms i.e. $g\left( x \right)={{e}^{x}},h\left( x \right)=f'\left( {{e}^{x}} \right)$
So, applying the formula we get
$\Rightarrow \dfrac{d}{dx}\left[ {{e}^{x}}.f'\left( {{e}^{x}} \right) \right]={{e}^{x}}\cdot f'\left( {{e}^{x}} \right)+\dfrac{d}{dx}\left[ f'\left( {{e}^{x}} \right) \right]\cdot {{e}^{x}}$ ……………………………(5)
As we can see in the above equation after plus sign there is the term which we have to derive using the formula given in equation (2). So, derivation of that particular term will be
$\dfrac{d}{dx}\left[ f'\left( {{e}^{x}} \right) \right]={{e}^{x}}\cdot f''\left( {{e}^{x}} \right)$
So, substituting this value back into equation (5), we get as
$\Rightarrow \dfrac{d}{dx}\left[ {{e}^{x}}.f'\left( {{e}^{x}} \right) \right]={{e}^{x}}\cdot f'\left( {{e}^{x}} \right)+{{e}^{x}}\cdot f''\left( {{e}^{x}} \right)\cdot {{e}^{x}}$
$\Rightarrow \dfrac{d}{dx}\left[ {{e}^{x}}.f'\left( {{e}^{x}} \right) \right]={{e}^{x}}\cdot f'\left( {{e}^{x}} \right)+{{e}^{2x}}\cdot f''\left( {{e}^{x}} \right)$ …………………………(6)
Now, putting value $x=1$ in equation (6) we get it as
$\Rightarrow \dfrac{d}{dx}\left[ {{e}^{x}}.f'\left( {{e}^{x}} \right) \right]=e\cdot f'\left( e \right)+{{e}^{2}}\cdot f''\left( e \right)$
On taking e common from the term, we get
$\Rightarrow \dfrac{d}{dx}\left[ {{e}^{x}}.f'\left( {{e}^{x}} \right) \right]=e\left( f'\left( e \right)+e\cdot f''\left( e \right) \right)$
Thus, we got the second order derivative as $e\left( f'\left( e \right)+e\cdot f''\left( e \right) \right)$
Hence, option (d) is the correct answer.

Note: Remember all the formulas and rules which are used in this type of sum. There are chances of mistakes when you consider $f\left( {{e}^{x}} \right)$ as $f\left( x \right)$ . So, first derivative will be $\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}$ and again taking second derivative will give answer as ${{e}^{x}}$ , if we use the property $\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}$ . So, this is not the correct method. You have to consider two different variables as shown in the given solution.