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Find the point(s) on the curve $y=6 x^{1 / 3}$ at which (a) the slope is $1 / 2$ (b) the tangent line is vertical.

(a) (64,24)(b) (0,0)

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.

03:00

tangent line equation

02:06

If $x y^{3}-y x^{3}=6$ is …

03:53

For each implicitly define…

01:18

Find an equation of the li…

02:08

Determine an equation of t…

The slope of the tangent l…

02:56

Find $y^{\prime}$ and the …

01:11

Tangent lines Find an equa…

All right, So we're examining this function six x to the one third power. And any time they ask you where the slope is equal to something, then they're saying, Okay, take the derivative, because that's your phrase for the slope. So bring that one third in front six times one third to be, too. And subtract one from that exponents. That would be negative. Two thirds Basically saying, when does that equation equal one half? Well, if I were doing this problem, I would take that derivative we just found and just rewrite it as you know, uh, negative expose six in the denominator, uh, three makes it the cube root of X squared. And then from here, just cross multiply. So I would have 42 attempts to us four and one times anything. Is that thing X squared? And so from here, I would just say Okay, well, undo the cube root Aiken Cube that before cube to 64 and tow. Undo the square. You have to square root the square root of 64 8. Uh, and so what I come up with is theirs. Uh, a couple of times that x. Yeah. This is my answer is not matching the direction. So, um, anyway, it should be plus or minus eight because if I go back to the original problem and I take But I guess if I plugged in eight into this problem Thank you, bro. Debate is two squared us four, and being the denomination to force reduces the one half some confidence that this is correct. But since I asked for the points, I need to go back to the original problem and come up with eight. With the cube. Root of eight is two times six will give you 12, but same thing with negative eight because the cube root of negative is negative two times six or giving negative 12 eso That's your answer to part A and your answer is a part B. I'll be in this color. Blue is basically saying, If we have a vertical tangent, then the denominator cannot equal zero. I guess would be easy to say it this way. Well, the only number squared and cubed rooted that will give you zeros of X equals zero. So back to the original problem zero to the one through power Still zero time 60 eso The answer to part B is 00 There you go

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