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Find the equation of the line perpendicular to the tangent line to the curve $$y=x^{3}-x+1$$ at $$x=2 .$$ Recall that lines are perpendicular if the product of their slopes is -1

$$y=-\frac{1}{11} x+\frac{79}{11}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

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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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I need to redo this problem because I made a tiny mistake. So you already have it mostly worked out. I'm just gonna underline as I go. So you're actually finding the normal line, which is the same problem as a tangent line problem where you need a point and a slope. We'll all you know is that X equals two. And we confined the slope for the Tanja line by finding the drift. Have been setting it equal to two. And I need to racist because it's the wrong value experience. Eso The first thing I did is I went to the original problem plugged in to and for all of these excess well, two times two times two is eight minus two is six plus one is seven. That makes sense. And then let's fix my mistake, because the derivative is bringing the three in front X squared. I forgot to subtract one from my exponents. I dragged over the X is one. The drift of other constant is zero. So now when I plug into into the derivative well, two squared is four times three is 12 minus one is 11. And so the slope of the normal line or the directions we want. The perpendicular line. It's the opposite reciprocal for negative reciprocal, which means you negate that number and put it. And the directions were that if you take the slope of the tangent times the slope of the perpendicular, the product will give you negative one. And that does so then our answer. I like to write in points so far, Um, where you write the slope X minus the x cornet plus the y coordinate. Or if you distribute that in and combine like terms. I was way off on that one. Um, yeah, Go back to my marker. There we go. Negative 1 11 x, you'd have plus to 11 plus seven. I would give you 79 11 if you like that. Answer instead. Both answers are acceptable. This one or this one.

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