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Differentiate the function.
$ T(z) = 2^z \log_2 z $
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00:31
Frank Lin
01:20
Doruk Isik
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 6
Derivatives of Logarithmic Functions
Derivatives
Differentiation
Oregon State University
Harvey Mudd College
Idaho State University
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.
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So for this problem, we're learning essentially how to take the derivatives of logarithmic functions. So what we're starting off in this case is a function t of Z. And it doesn't matter how we write this weaken right as TMZ ffx whatever, because just a function. Um but in this case, what we'll have is t of Z being equal to too busy times log based too, Zeke. So we're gonna want to use what's known as the product group for differentiation and how that's gonna look is we take the derivative of the first term that we're multiplying by. So to to the Z, we know that, uh, the derivative of to To the Z is going to be to To the Z I'm the natural log of to and then that will be multiplied by just log based too busy. We don't have to differentiate that at first. And then we add that with two to the scene times the derivative of this portion right here we know that the derivative of logarithmic function is going to be one over the U value. In this case, that would be easy times the natural log of the base value, which is to So that should be our answer right here. Then what we can do is simplify things further. You could either factor out the two Z. Um, that's one option or another thing that can be done is, um, simplifying things further. If we view this right here as the natural log of Z over the natural log of too, that's another way you could write it. If we do that, then this will cancel with this right here. Eso we would have that and then we can write to to the Z up top here. So this would be our final, um, mawr simplified form of it. We could also factor out of Tuesday if we wanted to. It would be too, too busy times natural log of Z plus one over z times, the natural rate of two. However, this could be an easier way of writing it. And then if we had t prime of Z, we would end up getting this exact same graph. Um but yeah, this ultimately just shows us how to dio a better job of differentiating logarithmic functions, which is crucial when dealing with applications of math which use a lot of rhythmic functions all the time.
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