Differentiate ln(x)/x

In order to differentiate something, you multiply the power that x is raised to by the coefficient (the number that you multiply x by). You then subtract 1 from the power. 

There are two different ways you could approach this problem. You could either use the product rule, or the quotient rule. I will only be running through the product rule as this is (usually) quicker and easier.

Remember that 1/x can be written as x-1

The product rule is that if y=u*v (where u and v are both functions of x), then


Let u=ln(x) and v=x-1 

So, dv/dx=-x-2 and du/dx=x-1 (I have explained why ln(x) differentiates to this in a seperate question)

Therefore, dy/dx=(x-1)(x-1) +(- x-2)ln(x)

This simplifies down to:

                                                           dy/dx= (1-ln(x))(x-2)

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