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# Find the derivative of the given term that is ${x^3}{e^x}$ ?

Last updated date: 23rd Feb 2024
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Hint: For differentiation you should know that after differentiating the term or equation given, power of variable is reduced by one, and differentiation of constant is always zero because derivative means measuring the change of a variable with respect to some quantity and as constant is always fixed so no change can be seen.

$\Rightarrow \dfrac{d}{{dx}}{x^3}{e^x}$
To solve this we have to use the product rule of differentiation which states that for any general equation say $x \times y$ differentiation by product rule can be given as:
$\Rightarrow \dfrac{d}{{dx}}x \times y = y\dfrac{d}{{dx}}x + x\dfrac{d}{{dx}}y$
$\Rightarrow \dfrac{d}{{dx}}{x^3} \times {e^x} = {e^x}\dfrac{d}{{dx}}{x^3} + {x^3}\dfrac{d}{{dx}}{e^x} \\ = {e^x}\left( {3{x^2}} \right) + {x^3} \times {e^x} \times \dfrac{d}{{dx}}\left( x \right) \\ = {e^x}\left( {3{x^2}} \right) + {x^3} \times {e^x} \times (1) \\ \Rightarrow {e^x}\left( {3{x^2}} \right) + {x^3} \times {e^x} \;$
So, the correct answer is “${e^x}\left( {3{x^2}} \right) + {x^3} \times {e^x}$”.