Question

Find the derivative of the function.

17. \(y = {x^2}{e^{ - 3x}}\)

Short answer

The derivative of the function is \(y' = x{e^{ - 3x}}\left( {2 - 3x} \right)\).

Step-by-step solution

The Chain Rule

For two functions, the chain rule is defined as:

\(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\)

Find the derivative of the function

When \(f\) and \(g\) are both differentiable, then the product ruleis denoted by;

\(\frac{d}{{dx}}\left( {f\left( x \right)g\left( x \right)} \right) = f\left( x \right)\frac{d}{{dx}}\left( {g\left( x \right)} \right) + g\left( x \right)\frac{d}{{dx}}\left( {f\left( x \right)} \right)\)

Use the product rule and chain rule to obtain the derivative of the function as shown below

\(\begin{aligned}y' &= \frac{d}{{dx}}\left( {{x^2}{e^{ - 3x}}} \right)\\& = {x^2}\frac{d}{{dx}}\left( {{e^{ - 3x}}} \right) + {e^{ - 3x}}\frac{d}{{dx}}\left( {{x^2}} \right)\\& = {x^2}\left( {{e^{ - 3x}}} \right)\frac{d}{{dx}}\left( { - 3x} \right) + {e^{ - 3x}}\left( {2x} \right)\\ &= {x^2}{e^{ - 3x}}\left( { - 3} \right) + {e^{ - 3x}}\left( {2x} \right)\\ &= {e^{ - 3x}}\left( { - 3{x^2} + 2x} \right)\\& = x{e^{ - 3x}}\left( {2 - 3x} \right)\end{aligned}\)

Thus, the derivative of the function is \(y' = x{e^{ - 3x}}\left( {2 - 3x} \right)\).