Differentiate ln(x)/x

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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

dy/dx=u(dv/dx)+v(du/dx)

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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