How do I differentiate

ln (2X^2 - 12X + Y^2 -10Y)

and the second differentiate it?

HELP ASAP is much appreciated

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wolframalpha dot com

When page be open in rectangle type:

derivative ln (2X^2 - 12X + Y^2 -10Y)

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After a few seconds , when you see result click option:

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second derivative is derivative of first derivative

on wolframalpha dot com type :

derivative 4 ( x - 3 ) / [ 2 x ^ 2 - 12 x + ( y -10 ) y ]

and click option =

then

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If this were

2x^2 - 12x + y^2 - 10y = 0
it would be an ellipse and we could do something with it
I will assume that much
secondly I will assume you want dy/dx

4x - 12 + 2y dy/dx - 10dy/dx = 0
dy/dx (2y-10) = -4x + 12
dy/dx = (-4x + 12)/(2y-10) = (2x-6)/(5-y)

d(dy)/(dx)^2 = ( (5-y)(2) - (2x-6)(-dy/dx) )/(5-y)^2

I then substituted dy/dx with (2x-6)/(5-y) and got

( 2(5-y)^2 - (2x-6)^2 )/(5-y)^3

you could expand that but it would look messier.

To differentiate the given expression, ln(2X^2 - 12X + Y^2 - 10Y), you need to apply the rules of differentiation with respect to the variables X and Y separately.

Step 1: Differentiation with respect to X
To differentiate with respect to X, treat Y as a constant and differentiate the expression as you would normally differentiate a function with respect to X.

d/dx[ln(2X^2 - 12X + Y^2 - 10Y)] = (1 / (2X^2 - 12X + Y^2 - 10Y)) * d/dx[2X^2 - 12X + Y^2 - 10Y]

Now, use the power rule of differentiation for each term:

d/dx[2X^2] = 4X
d/dx[-12X] = -12
d/dx[Y^2] = 0 (since Y is treated as a constant)
d/dx[-10Y] = 0 (since Y is treated as a constant)

Combining these derivative expressions, the first derivative with respect to X is:

d/dx[ln(2X^2 - 12X + Y^2 - 10Y)] = (4X - 12) / (2X^2 - 12X + Y^2 - 10Y)

Step 2: Differentiation with respect to Y
To differentiate with respect to Y, treat X as a constant and differentiate the expression as you would normally differentiate a function with respect to Y.

d/dy[ln(2X^2 - 12X + Y^2 - 10Y)] = (1 / (2X^2 - 12X + Y^2 - 10Y)) * d/dy[2X^2 - 12X + Y^2 - 10Y]

Now, use the power rule of differentiation for each term:

d/dy[2X^2] = 0 (since X is treated as a constant)
d/dy[-12X] = 0 (since X is treated as a constant)
d/dy[Y^2] = 2Y
d/dy[-10Y] = -10

Combining these derivative expressions, the first derivative with respect to Y is:

d/dy[ln(2X^2 - 12X + Y^2 - 10Y)] = (2Y - 10) / (2X^2 - 12X + Y^2 - 10Y)

So, the first partial derivatives with respect to X and Y are:

∂/∂X ln(2X^2 - 12X + Y^2 - 10Y) = (4X - 12) / (2X^2 - 12X + Y^2 - 10Y)

∂/∂Y ln(2X^2 - 12X + Y^2 - 10Y) = (2Y - 10) / (2X^2 - 12X + Y^2 - 10Y)

To find the second derivative, you'll need to differentiate these expressions again with respect to X and Y.