Next, we differentiate dxdy with respect to x to find dx2d2y:
dxd(1+4xy−4yy)
Using the quotient rule:
dx2d2y=(1+4xy)2(1+4xy)⋅dxd(−4yy)−(−4yy)⋅dxd(1+4xy)
First, we find dxd(−4yy):
dxd(−4yy)=−4(dxd(yy))
Using the product rule and chain rule:
dxd(yy)=y⋅dxdy+y⋅2y1⋅dxdy=y⋅dxdy+2yy⋅dxdy=(y+2yy)dxdy=23y⋅dxdy
So:
dxd(−4yy)=−4⋅23y⋅dxdy=−6y⋅dxdy
Next, we find dxd(1+4xy):
dxd(1+4xy)=4(y+x⋅2y1⋅dxdy)=4y+y2x⋅dxdy
Putting it all together:
dx2d2y=(1+4xy)2(1+4xy)(−6y⋅dxdy)−(−4yy)(4y+y2x⋅dxdy)
Simplifying each term:
=(1+4xy)2−6y⋅dxdy−24xy⋅dxdy−(−16y⋅y−8xy⋅dxdy)=(1+4xy)2−6y⋅dxdy−24xy⋅dxdy+16y2+8xy⋅dxdy=(1+4xy)2−6y⋅dxdy−16xy⋅dxdy+16y2
Finally, substitute dxdy back in to get the explicit formula for dx2d2y.
Would you like further details or have any questions?
How do we apply the chain rule in differentiation?
What is the quotient rule and how is it used in differentiation?
How can implicit differentiation help solve complex equations?
What are some common applications of second derivatives?
How do product and chain rules differ in their applications?
Tip: Always simplify expressions step by step to avoid errors in differentiation.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation Chain Rule Product Rule Quotient Rule Second Derivative