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Differentiate $\left(2 x+x^{4}\right)^{2}$ with respect to $x^{3} .$ Hint: let $y=\left(2 x+x^{4}\right)^{2}, u=x^{3}$ and use $\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}$.

$$\frac{4\left(2 x+x^{4}\right)\left(1+2 x^{3}\right)}{3 x^{2}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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So first we want to replace the equation. Um X cubed anywhere in ry equation with you. So we'll do that first to get replaced here which will be X cubed times X. Right? That's equal to X over four. But since this is now you you'll get two X. Close X. You to the second power. So we're going to do this in three steps one We're going to get we're going to have to use chain rule to use to get dy over do you? So you first want to find the derivative to lie with respect to you. So this is our variable. Everything else can be treated as a constant. And since we're using chain rule we're going to find the derivative of the inside first and then So the derivative of the inside with respect to you, we treat two X as a constant. So we're left with this term and we're left with X as the derivative of the inside. And now the outside of the equation that do it, if we can use power the power rule. So you have bring to down and you multiply that by the entire inside two x plus X. You To the one power. Now we want to find the U. two DX but the you want to find do you over D. X. So we use power rule here to find the derivative of X three because that is equal to you. So the drift of X. Of X. Three. Again using power rule is three X squared. Now this is our long version of the equation. So we can combine terms. So again this right here S. T. Y. Who do you? And this right here is do you over dx. So to combine terms we're going to have let's multiply this out first. So we have four X plus mhm Two X. You that quantity times X times three X squared. And these two combined Can be three x cubed Next. Our final step replacing you with X cubed again. Well, I circled the wrong thing there. So this year will become X cubed which will give us four X plus two X to the 4th Times three X. Cute. Which is a preservative.

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