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Determine the equation of the tangent line at the indicated $x$ -value.$$y=4 x+2 / x-1 ; \quad x=2$$

$$y=\frac{7}{2} x+1$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

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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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The whole premise of a tangent line is that you need a point in a slope and then you can use point slope form in order to come up with the answer. And then this problem all they give you is that excess too. And since they give us the notation of y equals out under this in D Y d. X. So the slope is found by finding the derivative and plugging in X equals two into that problem. So what's the equation that they give you? Why equals four X, what? Plus two over X. But I like to write his two X to the native first power, um, minus one. And as we plug in to and for all these, you know, here and here what we end up with this eight, uh two divided by two is one minus one eso pretty clearly that you get the answer of eight there. But the reason why I like to write in this form is then when I take the derivative, it makes perfect sense to me that that's for a derivative of four X. And then when you move this negative one in front of multiply and subtract one from your expert. It makes a little bit more sense. Thank anything else and the derivative of naked 10 So as I plug in to into this, remember, that means two squared in the denominator. So we're looking at four minus, um, to force or four minus one half, which is seven halves. Um, yes. So that's your derivative. Uh, because that's the derivative when X equals two, that's your slope. So what's your final answer? I like point. So for me, So I just write my answers. Why? It was seven halves the slope, x minus T X coordinates plus the Y coordinates. But some teachers will actually make you distribute. You know, I'm not that teacher, but maybe maybe you have that teacher where you say Okay, well, seven half times x seven has. When you distribute, you get negative. Seven plus eight, you get negative. Seven plus eight will give you one. There's that answer. Either one is acceptable

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