Questions: The Lambert function, W(x), is implicitly defined by the following equation: W(x) e^(W(x))=x Use implicit differentiation to find a formula for d/dx W(x). [Note: W(x) can be written as W ] W'= [There should not be any " x " terms in the answer.] Show that W'(x)=1/(e^W+x).

Solution Steps

To find the derivative of the Lambert function \( W(x) \) implicitly defined by \( W(x) e^{W(x)} = x \), we can use implicit differentiation. We differentiate both sides of the equation with respect to \( x \) and solve for \( \frac{dW}{dx} \).

1. Start with the equation \( W(x) e^{W(x)} = x \).
2. Differentiate both sides with respect to \( x \).
3. Use the product rule on the left-hand side.
4. Solve for \( \frac{dW}{dx} \).

We start with the equation defining the Lambert function: \[ W(x) e^{W(x)} = x \] To find \( \frac{dW}{dx} \), we differentiate both sides with respect to \( x \).

Using the product rule on the left-hand side, we have: \[ \frac{d}{dx}(W e^{W}) = \frac{d}{dx}(x) \] This gives us: \[ W e^{W} \frac{dW}{dx} + e^{W} \frac{dW}{dx} = 1 \]

Factoring out \( \frac{dW}{dx} \) from the left-hand side, we get: \[ \left(W e^{W} + e^{W}\right) \frac{dW}{dx} = 1 \] Thus, we can solve for \( \frac{dW}{dx} \): \[ \frac{dW}{dx} = \frac{1}{W e^{W} + e^{W}} = \frac{1}{e^{W}(W + 1)} \]

The expression simplifies to: \[ \frac{dW}{dx} = \frac{e^{-W}}{W + 1} \]

Final Answer