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Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $$ y = \tan \pi x $

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$=\pi \sec ^{2} \pi x$

00:29

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

00:50

Write the composite functi…

01:10

02:27

01:42

03:04

02:01

05:19

02:12

01:43

Write the composite func…

01:01

01:18

02:58

here we have a composite function y equals tangent of Pi X. Now there could be parentheses around the pie X. They didn't write them in in the book, but it means the same thing and having the parentheses there, it might make it a little easier to think about this problem. So we want to identify the inside function and the outside function. Let's go with what's inside the parentheses as their inside function g of X so g of X equals pi times x and then the outer function would be f of X equals tangent of X. Now we're going to find the derivative Do I. D. X. So we start by taking the derivative of the outside function, and the derivative of tangent is C can't squared. So we have c can't squared of pi X Still what was inside the original. Then we multiply by the derivative of the outside function. Our excuse me, the inside function and the derivative of the inside function would be the derivative of pi times X, So that would be pie. Okay, so we have the derivative and now, to simplify the answer, we might decide just to take this pie and right in front of the other part. So we have pie times, See? Can't squared of pi X.

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