Question

Find the 1000^th derivative of \(f\left( x \right) = x{e^{ - x}}\).

Short answer

The 1000^th derivative of the function is \({f^{\left( {1000} \right)}}\left( x \right) = \left( {x - 1000} \right){e^{ - x}}\).

Step-by-step solution

The Chain Rule

The chain rule is defined as:

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

By using Leibniz notation,

\(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\frac{{du}}{{dx}}\)

Determine the 1000th derivative of the function

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

\(\begin{array}{c}f'\left( x \right) = x\frac{d}{{dx}}\left( {{e^{ - x}}} \right) + {e^{ - x}}\frac{d}{{dx}}\left( x \right)\\ = - x{e^{ - x}} + {e^{ - x}}\\ = \left( {1 - x} \right){e^{ - x}}\end{array}\)

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

\(\begin{array}{c}f''\left( x \right) = \left( {1 - x} \right)\frac{d}{{dx}}\left( {{e^{ - x}}} \right) + {e^{ - x}}\frac{d}{{dx}}\left( {1 - x} \right)\\ = \left( {1 - x} \right)\left( { - {e^{ - x}}} \right) - {e^{ - x}}\end{array}\)

Use the chain rule to evaluate a few derivatives and the 1000^th derivative of the function as shown below:

\(\begin{array}{c}f'''\left( x \right) = \left( {1 - x} \right)\frac{d}{{dx}}\left( { - {e^{ - x}}} \right) - {e^{ - x}}\frac{d}{{dx}}\left( {1 - x} \right) - \frac{d}{{dx}}\left( {{e^{ - x}}} \right)\\ = \left( {1 - x} \right){e^{ - x}} + {e^{ - x}} + {e^{ - x}}\\ = \left( {3 - x} \right){e^{ - x}}\\{f^{\left( 4 \right)}}\left( x \right) = \left( {x - 4} \right){e^{ - x}}\\ \vdots \\{f^{\left( {1000} \right)}}\left( x \right) = \left( {x - 1000} \right){e^{ - x}}\end{array}\)

Thus, the 1000^th derivative of the function is \({f^{\left( {1000} \right)}}\left( x \right) = \left( {x - 1000} \right){e^{ - x}}\).