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Find the derivative of the function.$ G(x) = 4^{C/x} $

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00:23

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

University of Nottingham

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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in this problem, we have an exponential function. So as a reminder, if you want to take the derivative of an exponential function like four to the X, it's going to be four to the X Times natural log. Four. So then, if we make it a composite function and we have four race to a function, it's going to be four. Race to that function times Natural log four. That's the derivative of the outside, multiplied by F prime of X. That's the derivative of the inside. So for this function will follow that rule. And first I'm going to rewrite it as four raised to the power See X to the negative one. Any time I have something divided by X, I think of it as X to the negative first power. Okay, so for the derivative, according to this exponential function derivative rule, we would have four race to the sea X to the negative one multiplied by natural log four multiplied by the derivative of C X to the negative one. And that would be negative one time See X to the negative too. So now what we need to do is just clean this up a little bit. So when I write four c x to the negative one, I'm gonna put that back into the fraction notation four to the sea over X Power times Natural log for times negative. See, In fact, I might just write that negative See in the front of my answer as we typically put a constant in the front and then we'll take this X to the negative. Second power will bring it down to the bottom as X squared.

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