Question

implicit differentiation dy/dx of(cos πx + sin πy)4 = 67

Answer 1

Is the 4 suppose to be an exponent, like this?
\[\huge (\cos \pi x+\sin \pi y)^4=67\]

Answer 2

use the chain rule :)

Answer 3

1. chain rule: \[\frac{ d }{ dx } a^n = na^{n-1}\]
2. differentiate with respect to x: \[\frac{ dy }{ dx }\]
3. use algebra to rearrange and solve for dy/dx
Your final answer should be in terms of only x and y.

Answer 4

Using the tips from @zepdrix and @abb0t:
Differentiate both sides:\[4(\cos \pi x + \sin \pi y)^3(-\pi \sin \pi x + \pi \cos \pi y \frac{ dy }{ dx })=0\]If you divide by 4pi you get:\[(\cos \pi x + \sin \pi y)^3(\frac{ dy }{ dx } \cos \pi y - \sin \pi x)=0\]A product is zero if one or more factors are zero:\[\cos \pi x + \sin \pi y = 0\]or\[\frac{ dy }{ dx }(\cos \pi y - \sin \pi x)=0\]This last one leads to:\[\frac{ dy }{ dx }=\frac{ 1 }{ \cos \pi y - \sin \pi x }\]