Did you know that finding derivatives of logarithmic functions has the same steps as used for exponential functions — just in reverse order and with the opposite operation?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
It’s true!
But why?
It’s because logarithmic functions are inverses of exponential functions!
Log Equation
And there are some important log properties, listed below, that allow us to simplify and solve logarithmic functions that are pivotal to our successful ability to simplify derivatives.
Properties Of Log Functions
Exponential Vs Logarithmic Derivatives
Alright, so now we’re ready to look at how we calculate the derivative of a logarithmic function, but before we do, let’s quickly review our 3 steps for differentiating an exponential function.
Why?
Since exponential functions and logarithmic functions are so similar, then it stands to reason that their derivatives will be equal as well.
Steps for differentiating an exponential function:
- Rewrite.
- Multiply by the natural log of the base.
- Multiply by the derivative of the exponent.
Derivative Of Exponential
So, now we’re going to learn the steps for differentiating logarithmic functions:
- Take the derivative of the function.
- Divide by the product of the natural log of the base and the rewritten function.
Log Derivative Formula
Did you notice something amazing?
These three steps are in reverse order from the steps for differentiating an exponential function, and instead of multiplication, we use division.
Why?
Because they’re inverses, so they “undo” each other (i.e., division is the opposite of multiplication)!
Consequently, log rules and exponential rules are very similar.
Derivatives Of Logs
Okay, so let’s find the derivative, using our new derivative rule, for the following logarithmic functions.
How To Differentiate Logarithmic Functions
Throughout our lesson, we will review our properties of logarithm and work through multiple examples of how to calculate and simplify derivatives of logarithmic functions,
Let’s get to it!
Video Tutorial w/ Full Lesson & Detailed Examples (Video)
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