How do you find the slope of a tangent line to the graph of the function #3xy-2x+3y^2=5# at (2,1)?

Answer 1

Please the explanation for steps leading to the slope, #m = -1/12#

Given: #3xy - 2x + 3y^2 = 5#
The steps are: 1. Implicitly differentiate the equation. 2. Obtain the slope of the tangent line by evaluating the derivative at the point #(2, 1)#

I will differentiate each term, separately.

1.1 Use the product rule for term 1:

#(uv)' = u'v + uv'#
let #u = 3x#, then #u' = 3, v = y and v' = dy/dx#
#(uv)' = 3y + 3xdy/dx#

1.2 Use the power rule on term 2:

#(d(-2x))/dx = -2#

1.3 Use the chain rule on term 3:

#(d(3y^2))/dx = 6ydy/dx#

1.4 Term 4 is a constant, 5, the derivative is zero.

Put these back into the equation:

#3y + 3xdy/dx - 2 + 6ydy/dx = 0#
Move the terms without #dy/dx# to the right:
#3xdy/dx + 6ydy/dx = 2 - 3y#
Factor out #dy/dx# on the left:
#(3x + 6y)dy/dx = 2 - 3y#
Divide both sides by #(3x + 6y)#:
#dy/dx = (2 - 3y)/(3x + 6y)#
The slope, m, of the tangent line is the above evaluated at the point #(2,1)#:
#m = (2 - 3(1))/(3(2) + 6(1))#
#m = -1/12#
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Answer 2

To find the slope of a tangent line to the graph of a function at a specific point, we can use the derivative. First, we need to find the derivative of the given function with respect to x. Then, we can substitute the x-coordinate of the given point into the derivative to find the slope of the tangent line. In this case, the derivative of the function 3xy - 2x + 3y^2 = 5 with respect to x is 3y - 2 + 6xy. Substituting the x-coordinate 2 into the derivative, we get the slope of the tangent line at (2,1) as 3y - 2 + 6xy = 3(1) - 2 + 6(2)(1) = 3 - 2 + 12 = 13. Therefore, the slope of the tangent line to the graph of the function 3xy - 2x + 3y^2 = 5 at (2,1) is 13.

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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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