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# Differentiate with respect to x: $3{{x}^{2}}-{{e}^{-3x}}+\sec x$.  Hint: We will be using the concept of differential calculus to solve the problem. We will be using chain rule of differentiation to solve the problem.

Now, we have to differentiate $3{{x}^{2}}-{{e}^{-3x}}+\sec x$ with respect to x.
So, we let,
$f\left( x \right)=3{{x}^{2}}-{{e}^{-3x}}+\sec x$
Now, we know that if $f\left( x \right)=g\left( x \right)+h\left( x \right)$ then,
$\dfrac{d}{dx}f\left( x \right)=\dfrac{d}{dx}g\left( x \right)+\dfrac{d}{dx}h\left( x \right)$
Therefore, we have,
$\dfrac{d}{dx}f\left( x \right)=\dfrac{d}{dx}\left( 3{{x}^{2}} \right)-\dfrac{d}{dx}\left( {{e}^{-3x}} \right)+\dfrac{d}{dx}\left( \sec x \right)$
Now, we know that,
$\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$
Now, we know that according to chain rule,
\begin{align} & \dfrac{d}{dx}\left( {{e}^{-ax}} \right)={{e}^{-ax}}\times \dfrac{d}{dx}\left( -ax \right) \\ & \dfrac{d}{dx}\left( \sec x \right)=\sec x\tan x \\ \end{align}
So, we have,
\begin{align} & \dfrac{d}{dx}f\left( x \right)=3\times 2x-\left( -3 \right){{e}^{-3x}}+\sec x\tan x \\ & =6x+3{{e}^{-3x}}+\sec x\tan x \\ \end{align}
Therefore, we have the differentiation of $3{{x}^{2}}-{{e}^{-3x}}+\sec x$ with respect to x as,
$6x+3{{e}^{-3x}}+\sec x\tan x$

Note: To solve these type of questions it is important to remember that if $f\left( x \right)=g\left( x \right)+h\left( x \right)$ then $f'\left( x \right)=g'\left( x \right)+h'\left( x \right)$. Also, it should be noted $\dfrac{d}{dx}\left( {{e}^{-3x}} \right)$ have been found by using chain rule.
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