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Determine the equation of the tangent line at the given $x$ -coordinate.$$f(x)=e^{-x}, x=0$$

$y=-x+1$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

Oregon State University

McMaster University

University of Michigan - Ann Arbor

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Okay. Any time you're asked to find the equation of a tangent line, all you need is a point and a slope. But now they give you the X coordinate for each of those, Um, so they already tell you the X corner. And the problem is zero. They don't tell you the y coordinate. And then, uh, pretty much the slope is the derivative when x zero. So that's all you have to find out. And when they gave you the equation for X ffx is e to the negative x. Well, it makes perfect sense when you plug zero in for this X that e to the zero power equals one and most are, I would say, every calculus. Students should know that whereas the derivative first find the derivative where the derivative of E to the negative X is either the negative x times, the derivative of negative X, which is negative one. So I'm going to take a negative one in front because multiplication is communicative. So now when I plug in zero for that X, you still get either the zero powers of one. But then don't forget about that negative in front. So it's really nice about this is, uh, actually when x zero that gave me the y intercept as well. And this is the slope. So my answer for the tangent lines. Negative one x plus one, and we're done. If you don't want to write that negative one, just make sure you have a negative in there. So there you go.

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