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(a) Find an equation of the tangent line to the curve $ y = 2/(1 + e^{-x}) $ at the point (0, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

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00:48

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Missouri State University

University of Nottingham

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.

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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(a) Find an equation of th…

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a. Find an equation of the…

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01:23

for this problem were given the function y equals 2/1 plus e to the negative X, and we want to find the equation of the tangent line at the 0.1 So remember to get the slope of the tangent line, you find the derivative. So we're going to use the quotient rule to find the derivative of the function. So what we have here is the bottom times, the derivative of the top, minus the top times, a derivative of the bottom. And we use the chain rule to find the derivative of E to the negative X. And then that's over the bottom squared. Next, we want to simplify that, and we want to plug in the number zero for X because we're finding the derivative at the 0.1 So a substitute in zero everywhere we have an X and remember that each of the zero is one, so that simplifies things quite a bit. And that gives us negative two times one times negative, 1/2 squared that simplifies to be 2/4. So that's 1/2. And remember, that is the slope of the tangent line. So now that we have the slope and we have the 0.1 We can use our point slope form of the equation of a line. Why minus y one equals M times X minus X one. We can substitute our slope in there and our 0.1 and that will give us the equation of the tangent line, distribute the 1/2 and then add one to both sides and we have y equals 1/2 X plus one. So that takes care of part A and then for part B. What we want to do is illustrate by graphing the curve and the Tanja line on the same screen. So we grab a calculator, we go to y equals and we type in. Our function is why one and we type in our tangent line is why too. All right now we graph and I'm using zoom decimal number four and the blue one is the curve, and the red one is a tangent line. And here we have the cursor at the 10.1 so we can see that that line does appear to be tangent to the curve. At that point

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