# Computing Derivatives

# Derivative of *ln* x

If we didn't already know the derivative of *ln* *x*, we could figure it out using the chain rule.

We know that

*e*^{{ln x}} = *x*.

Take the derivative of each side of this equation.

The derivative of *x* is 1.

To find the derivative of *e*^{{ ln x}} we need to use the chain rule. The outside function is* e*^{{□}} and its derivative is also *e*^{{□}.}

The inside function is *ln* *x*. Since we don't yet know the derivative of *ln* *x* (at least, we're pretending we don't) we'll write its derivative as (*ln* *x*)'.

The chain rule says

(*e*^{{ ln x}})' = *e*^{{ ln x}} × (*ln* *x*)'

Since *e*^{{ln x}} = *x*, we can simplify this to

(*e*^{{ ln x}})' = *x* × (*ln* *x*)'

Now return to the equation

*e*^{{ln x}} = *x*.

The derivative of the right-hand side is 1, and the derivative of the left-hand side is *x* × (*ln* *x*)', therefore

*x* × (*ln* *x*)' = 1.

Dividing both sides by *x*, we find