How do you find the slope of the tangent line to the curve #6x^2 - 2xy + 3y^3 = 216# at the point 6,0?

Answer 1

Use implicit differentiation with respect to #x# to get: #12x-2y-2x\cdot dy/dx+9y^2\cdot dy/dx=0# (note that the Chain Rule is used). Now solve for #dy/dx# to get #dy/dx=(2y-12x)/(-2x+9y^2)#. Finally, plug in the point #(x,y)=(6,0)# to get #dy/dx|_{(x,y)=(6,0)}=\frac{-72}{-12}=6#.

By the way, it is worth confirming that #(x,y)=(6,0)# is on the original curve: #6\cdot 6^{2}-2\cdot 6\cdot 0+3\cdot 0^{3}=216#.

Also, we are assuming #y# is a function of #x# when we differentiate the original equation. In fact, you may find it more helpful to rewrite the original equation as #6x^2-2xf(x)+3(f(x))^3=216# and then, after differentiation, write #12x-2f(x)-2xf'(x)+9(f(x))^2f'(x)=0#. Now solve for #f'(x)# to get #f'(x)=(2f(x)-12x)/(-2x+9(f(x))^2)# and plug in #x=6# and #f(6)=0# as above.

Here's a picture of the situation in this problem. The tangent line at the given point is shown as well (dashed).

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Answer 2

To find the slope of the tangent line to the curve (6x^2 - 2xy + 3y^3 = 216) at the point ( (6,0) ), follow these steps:

  1. Differentiate the given equation implicitly with respect to (x) to find ( \frac{dy}{dx} ).
  2. Evaluate ( \frac{dy}{dx} ) at the point ( (6,0) ) to find the slope of the tangent line.
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Answer 3

To find the slope of the tangent line to the curve 6x^2 - 2xy + 3y^3 = 216 at the point (6,0), first, differentiate the equation implicitly with respect to x. Then, substitute the x-coordinate (6) and y-coordinate (0) into the resulting expression. Solve for dy/dx to find the slope of the tangent line at the given point.

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Answer from HIX Tutor

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.

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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