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Differentiate the function.

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

$\frac{1}{2 x \ln 10}$

00:24

Frank L.

00:50

Doruk I.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 6

Derivatives of Logarithmic Functions

Derivatives

Differentiation

Oregon State University

Harvey Mudd College

Baylor University

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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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