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Differentiate the function.

$ f(x) = \log_10 \sqrt x $

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00:24

Frank Lin

00:50

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Missouri State University

Oregon State University

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

01:51

find the second derivative…

01:00

Differentiate the function…

00:57

01:08

01:28

02:05

So for this problem, what we're gonna want to dio we have log rhythm based derivatives that we're doing In this case, what we have is f of X being equal Thio Log based 10. The log based 10 is implied so we can just call the log of the square root of X. Now we know that in this case will apply the rule. We know that the log based be of you is equal to one over you times the natural log of B and then times the derivative of you so it will end up getting here for the derivative is one over you. In this case, the you is the square root of acts, um, times the natural log of B being the base of the natural log of 10. And lastly, we want to multiply this times the derivative of the U portion or the square root of X, and that's gonna ultimately just end up giving us one over the square root 1/2 times the square. Divac's both all this in mind. We can combine everything we see that this squared of acts time two squared of acts will just give us X and then to we will have out in front, so we'll get it to x times the natural 10 with one over that. This will be our final graph of the function. And we see that if this is our our FX right here with base 10, we see that this would be ultimately the same result right here. If we were to graph f prime of X, we see we end up getting that portion. That's because the log we can't have a negative value for X, so it's just going to be the positive portion of this graph. That's the actual derivative. It's not differential in the negative values of X.

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