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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second derivative is derivative of first derivative
on wolframalpha dot com type :
derivative 4 ( x - 3 ) / [ 2 x ^ 2 - 12 x + ( y -10 ) y ]
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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.