Pierce Mccoy
06/14/2024 · Elementary School
(a) Find \( \frac{\mathrm{d}}{\mathrm{d} x}\left(x \mathrm{e}^{-2 x}\right) \)
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Tutor-Verified Answer
Step-by-step Solution
To find the derivative of the function \( f(x) = x \mathrm{e}^{-2x} \), we will use the product rule. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:
\[
\frac{\mathrm{d}}{\mathrm{d} x}(u v) = u' v + u v'
\]
In this case, we can let:
- \( u(x) = x \)
- \( v(x) = \mathrm{e}^{-2x} \)
Now, we need to find the derivatives \( u' \) and \( v' \):
1. The derivative of \( u(x) = x \) is:
\[
u' = 1
\]
2. The derivative of \( v(x) = \mathrm{e}^{-2x} \) can be found using the chain rule:
\[
v' = \frac{\mathrm{d}}{\mathrm{d} x}(\mathrm{e}^{-2x}) = \mathrm{e}^{-2x} \cdot (-2) = -2 \mathrm{e}^{-2x}
\]
Now we can apply the product rule:
\[
\frac{\mathrm{d}}{\mathrm{d} x}(x \mathrm{e}^{-2x}) = u' v + u v' = (1)(\mathrm{e}^{-2x}) + (x)(-2 \mathrm{e}^{-2x})
\]
This simplifies to:
\[
\mathrm{e}^{-2x} - 2x \mathrm{e}^{-2x}
\]
We can factor out \( \mathrm{e}^{-2x} \):
\[
\frac{\mathrm{d}}{\mathrm{d} x}(x \mathrm{e}^{-2x}) = \mathrm{e}^{-2x}(1 - 2x)
\]
Thus, the final result is:
\[
\frac{\mathrm{d}}{\mathrm{d} x}(x \mathrm{e}^{-2x}) = \mathrm{e}^{-2x}(1 - 2x)
\]
Quick Answer
The derivative of \( x \mathrm{e}^{-2x} \) is \( \mathrm{e}^{-2x}(1 - 2x) \).
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