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Find the differential of each function.(a) $ y = xe^{4x} $(b) $ y = \sqrt {1 -t^4} $

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Derivatives

Differentiation

Harvey Mudd College

Baylor University

University of Nottingham

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.

02:48

Find the differential $d y…

00:56

Find the differential of t…

01:33

01:22

04:19

00:49

Find the differential of e…

01:27

02:30

okay. Part A The derivative using the product rule. We have acts times You did the negative for ox Natural order of E. We know that we're taking the derivative of negative four acts. So in this case, this simplifies to negative for X E to the negative or ax plus e to the night of four ax using the chain rule and factoring we end up with Okay, Part B. We know that we have one nice t to the fourth to the 1/2. Therefore we have, if you have tea, is one nice to the fourth than we know that we could have 1/2 you to the negative. 1/2 comes negative for t cubed, which is negative to t cubed over the square root of one minus t to the fourth.

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