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Chapter 2: Problem 14

Find \(d y / d x\) by implicit differentiation. $$ (\sin \pi x+\cos \pi y)^{2}=2 $$

Short Answer

Expert verified
\(dy/dx = -\frac{\cos \pi x}{\sin \pi y}\)

Step by step solution

01

Differentiate both sides of the equation

Applying the power rule and the chain rule, differentiate the left hand side of the equation with respect to \(x\). For the right hand side that is a constant, its derivative is zero. So, get \((2 (\sin \pi x+\cos \pi y)) (\pi \cos \pi x - \pi \sin \pi y \, dy/dx)=0\)
02

Simplify the equation

Simplify the equation and isolate the \(dy/dx\) term in the equation. This gives \(dy/dx = \frac{(2 \pi \cos \pi x (\sin \pi x+\cos \pi y))}{(2 \pi \sin \pi y (\sin \pi x+\cos \pi y))}\)
03

Cancel common terms to get the final solution

Cancel the common terms and this simplifies the expression for \(dy/dx\) to \-\(\frac{\cos \pi x}{\sin \pi y}\)

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