dy/dx by implicit differentiation is cos(πx)/sin(πy)
<h3>How to find dy/dx by implicit differentiation?</h3>
Since we have the equation
(sin(πx) + cos(πy)⁸ = 17, to find dy/dx, we differentiate implicitly.
So, [(sin(πx) + cos(πy)⁸ = 17]
d[(sin(πx) + cos(πy)⁸]/dx = d17/dx
d[(sin(πx) + cos(πy)⁸]/dx = 0
Let sin(πx) + cos(πy) = u
So, du⁸/dx = 0
du⁸/du × du/dx = 0
Since,
- du/dx = d[sin(πx) + cos(πy)]/dx
= dsin(πx)/dx + dcos(πy)/dx
= dsin(πx)/dx + (dcos(πy)/dy × dy/dx)
= πcos(πx) - πsin(πy) × dy/dx
So, du⁸/dx = 0
du⁸/du × du/dx = 0
8u⁷ × [ πcos(πx) - πsin(πy) × dy/dx] = 0
8[(sin(πx) + cos(πy)]⁷ × (πcos(πx) - πsin(πy) × dy/dx) = 0
Since 8[(sin(πx) + cos(πy)]⁷ ≠ 0
(πcos(πx) - πsin(πy) × dy/dx) = 0
πcos(πx) = πsin(πy) × dy/dx
dy/dx = πcos(πx)/πsin(πy)
dy/dx = cos(πx)/sin(πy)
So, dy/dx by implicit differentiation is cos(πx)/sin(πy)
Learn more about implicit differentiation here:
brainly.com/question/25081524
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