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## Main Question or Discussion Point

Here was my thinking for differentiation (which, by the way, is wrong):

By the definition of the function, the following equations are equal:

$$W(xe^x)=x$$

By the chain rule and product rule:

$$\frac{dW}{dx}( e^x+xe^x ) =1$$

$$\frac{dW}{dx}=(e^x+xe^x)^{-1}$$

What is the error here? What is the correct way to differentiate it? Also, how would I integrate it?

P.S. Why is the fraction bar in the third equation bolder than in the second equation? They are typesetted the same way...

By the definition of the function, the following equations are equal:

$$W(xe^x)=x$$

By the chain rule and product rule:

$$\frac{dW}{dx}( e^x+xe^x ) =1$$

$$\frac{dW}{dx}=(e^x+xe^x)^{-1}$$

What is the error here? What is the correct way to differentiate it? Also, how would I integrate it?

P.S. Why is the fraction bar in the third equation bolder than in the second equation? They are typesetted the same way...