Find by implicit differentiation.

Question

Find  by implicit differentiation.

Quizplus · Accepted Answer

Implicit differentiation involves finding the derivative of an equation with respect to a variable that is not explicitly stated. In this case, we can use implicit differentiation to find the derivative of the given equation with respect to x.

Starting with the given equation: 11eb12b0_aa07_95f4_b394_9f6b36592950_TB7866_11

We can begin by applying the chain rule:

d/dx (11eb12b0_aa07_95f4_b394_9f6b36592950_TB7866_11) = d/dx (y)

The derivative of the constant term is zero, so we can drop it:

d/dx (y) = d/dx (11eb12b0_aa07_95f4_b394_9f6b36592950_TB7866_11)

Next, we need to differentiate each term in the equation with respect to x:

d/dx (x^2 + y^2) = d/dx (3x^2 - 3y^2)

Using the power rule for differentiation:

2x + 2y * dy/dx = 6x - 6y * dy/dx

Now we can solve for dy/dx:

2y * dy/dx + 6y * dy/dx = 6x - 2x

8y * dy/dx = 4x

dy/dx = 1/2 * x/y

Substituting in the given point (x=3, y=4), we get:

dy/dx = 1/2 * 3/4 = 3/8

Therefore, the equation of the tangent line at the point (3,4) is:

y - 4 = 3/8 * (x - 3)

Simplifying:

8y - 32 = 3x - 9

3x - 8y = 23

Converting to the given format, we get:

11eb12b0_aa07_95f6_b394_477713bb6340_TB7866_11

Find by Implicit Differentiation