Have you ever run across a derivative that you couldn’t take it directly, and you felt like throwing up your hands in frustration?


Jenn, Founder hunterriverpei.com®, 15+ Years suffer (Licensed & Certified Teacher)

Don’t issue — we’ve all been there.

You are watching: Derivative of ln(1+x)

In fact, over there are plenty of instances as soon as direct differentiation (i.e., power rule, chain rule, quotient rule, etc.) is together a beast that we must look for a various method.

That’s when the logarithmic differentiation comes right into play!

We use this method when the plain rules that differentiation it seems to be ~ algebraically complicated.

For instance, which perform you think would certainly be simpler to differentiate?


Log – Condensed Vs broadened Form

Now, at first glance, you might be tempted to say the left side since it’s every one term.

But after additional inspection, i hope you will acknowledge that it’s the duty on the right because it’s only a streamlined version the the left. That also makes that much easier to calculate!

Here’s why.


Derivative – log Condensed Vs increased Form

By using the properties of logarithms, the appropriate side came to be much more manageable.

So, as soon as it’s about to obtain complicated, pull out those log in properties.


Properties Of log Functions

Logarithmic Differentiation Steps

But what happens if we are provided a function that isn’t logarithmic — deserve to we still use this method to make acquisition derivatives easier?

You bet.

Just monitor the 5 steps below:

Take the herbal log the both sides.Use log in properties to simplify the equations.Differentiate both sides making use of implicit differentiation and other derivative rules.Solve for dy/dx.Replace y through f(x).


For instance, finding the derivative that the duty below would be incredibly complicated if we were separating directly, but if we use our measures for logarithmic differentiation, climate the procedure becomes lot easier.

Take a look!


Logarithmic Differentiation – Example

So, all we did to be take the organic logarithm the both political parties of the equation, apply the logarithm nature to different terms, distinguish implicitly, and simplify.


While logarithmic differentiation can help us with algebraically tricky questions, this technique’s actual power is as soon as we are given an expression where one change is raised to one more variable — and the common rules for derivatives don’t apply.

How do you take it a derivative that a variable raised to a variable?

Function increased To A Function

Rewrite the equation so that the variables room no longer exponents with the help of logarithmic differentiation.

For example, mean we space asked to discover the following function’s derivative.

Function Sin X raised To 2x Power

Notice the the variable, x, shows up as the base and also the exponent. Therefore, the usual rules of straight differentiation don’t apply.

Consequently, we need to follow the procedures for logarithmic differentiation.

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Logarithmic Differentiation the Trig Functions

It’s the simple!

Together we will look at 5 questions involving polynomials, trig, exponentials, and of course, log in functions, as we learn how to use logarithmic differentiation with ease.