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While we have stated the chain rule, for the most part we examined the special case of the generalized power rule, as we have been dealing only with algebraic type functions. Later on in the text, we shall consider other kinds of functions which will utilize the chain rule. we anticipate other applications of the chain rule.Suppose $y=T(x)$ where $\frac{d}{d x}(T(x))=S^{2}(x),$ determine $\frac{d}{d x}\left(T\left(x^{2}\right)\right).$

$$2 x S^{2}\left(x^{2}\right)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

Baylor 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.

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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While we have stated the c…

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The generalized power rule…

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Let $w=f(x, y)$ be a funct…

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Use a tree diagram to writ…

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Let $z=f(t) g(t) .$ Use th…

All right. So we're gonna use the chain room. This problem and way define the derivative of some function. Uh, t of X is equal to capital s squared of x. Another way of writing that is like the quantity of SFX squared. Uh, so that's like a little thought bubble in kiss you, in case that's new for you. But anyway, the main premise here is that, um, the derivative of this function. Yeah, the ex stays the same, but But then, if you think of it this way, is it's a changeable, because once you do this function, so the derivative of this, this function, then you multiply by the drink. So as we get to the second part, which is the chief of X square, oops. Forgot to write squared. The function stays the same, so it's still s squared of X squared. But then you have to multiply by the derivative of X squared, which would be two X. You could leave your answer like that, or you could write it to x times capital s squared of X squared. This is the answer we're looking for. Now. If anybody is curious what they're trying to get out with this problem, I believe, is the derivative of tangent Tangent of X, Because I use the next there that's actually equal to seek in squared of X. So it kind of leading you down the path and, uh so you can see that the derivative of Tanja becomes seeking squared. But then, with the changeable, the X stays the same unless the function is X squared, then you have to take the derivative of the inside, which is two X. Anyway, this is the correct answer, and hopefully I've led you down a path that this makes sense.

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