Question

What is the derivative of $\left( {{x}^{2}} \right)\left( {{e}^{x}} \right)$ ?

Answer 1

Here in this question we have been asked to find the derivative of $\left( {{x}^{2}} \right)\left( {{e}^{x}} \right)$ for answering this question we will use the concept of derivatives for product of two different functions $u$ and $v$ of $x$ will be given as $\dfrac{d}{dx}\left( uv \right)=u\left( \dfrac{dv}{dx} \right)+v\left( \dfrac{du}{dx} \right)$.

Complete step by step answer:
Now answering the question we have been asked to find the derivative of $\left( {{x}^{2}} \right)\left( {{e}^{x}} \right)$ .
From the basic concepts of derivatives we know that the derivation of product of two different functions $u$ and $v$ of $x$ will be given as $\dfrac{d}{dx}\left( uv \right)=u\left( \dfrac{dv}{dx} \right)+v\left( \dfrac{du}{dx} \right)$ .
We can write the given expression mathematically as $\dfrac{d}{dx}\left( {{x}^{2}} \right)\left( {{e}^{x}} \right)$ .
By using the concept of derivation of product of two functions in the given case we will have $\Rightarrow {{x}^{2}}\dfrac{d}{dx}{{e}^{x}}+{{e}^{x}}\dfrac{d}{dx}{{x}^{2}}$ .
We know that there are some standard formula in derivations some of them are useful in the process of answering this question which are $\dfrac{d}{dx}{{e}^{x}}={{e}^{x}}$ and $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ .
By using the above described formula in the given case we will have ${{x}^{2}}\left( {{e}^{x}} \right)+{{e}^{x}}\left( 2x \right)$ .
Now by further simplifying this expression we will have $x{{e}^{x}}\left( x+2 \right)$ .

Therefore we can conclude that the derivative of $\left( {{x}^{2}} \right)\left( {{e}^{x}} \right)$ will be given as $x{{e}^{x}}\left( x+2 \right)$

Note: In the process of answering questions of this type we should be very sure with the formula that we are going to apply in between the steps. Similarly we have many other derivative formula given as $\dfrac{d}{dx}{{e}^{ax}}=a{{e}^{ax}}$ and $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$ . So it is very advisable to remember all the basic formulae related to the concept of derivations and integrations. Integration is the inverse operation of derivation.