# What is the slope of the tangent line of #r=2theta^2-3thetacos(2theta-(pi)/3)# at #theta=(-5pi)/3#?

Derivative with Polar Coordinates is

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To find the slope of the tangent line at a given point on a polar curve, you can use the formula ( \frac{dy}{dx} = \frac{r' \cdot \sin(\theta) + r \cdot \cos(\theta)}{r' \cdot \cos(\theta) - r \cdot \sin(\theta)} ), where ( r' ) is the derivative of ( r ) with respect to ( \theta ). Plugging in the values for ( r ), ( r' ), and ( \theta ) given in the question, we get:

( r = 2\theta^2 - 3\theta \cos(2\theta - \frac{\pi}{3}) ) ( r' = 4\theta - 3(2\theta - \frac{\pi}{3}) \sin(2\theta - \frac{\pi}{3}) )

Now substitute ( \theta = -\frac{5\pi}{3} ) into these equations to find ( r ) and ( r' ), then plug them into the slope formula to calculate the slope of the tangent line at that point.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- What is the distance between the following polar coordinates?: # (6,pi/3), (0,pi/2) #
- How do you find the polar coordinates of #(-4,0)# ?
- What is the Cartesian form of #(-1,(14pi)/3))#?

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