Find dy / dx by implicit differentiation. sin⁡x+2cos⁡2y=1

Question

Find dy / dx by implicit differentiation.  sin⁡x+2cos⁡2y=1

Derrick · Accepted Answer

Step 1Implicit differentiation is performed when the dependent variable cannot be explicitly written as a function of the independent variable. To perform implicit differentiation use ddxf(y)=f′(y)dydx.Derivative of sin⁡x is equal to cos⁡x. Derivative of cos⁡ax is equal to −a sin⁡ax. The derivative of constant is equal to 0. Use the property of derivatives (af(x))′=af&#039;(x) where a is a constant.Step 2Given equation is sin⁡x+2cos⁡2y=1. Use information from step 1 to compute dydx.sin⁡x+2cos⁡2y=1ddx(sin⁡x+2cos⁡2y)=ddx(1)ddxsin⁡x+ddx(2cos⁡2y)=0cos⁡x+2ddxcos⁡2y=0cos⁡x−2(2sin⁡2y)=0cos⁡x−4sin⁡2ydydx=04sin⁡2ydydx=cos⁡xdydx=cos⁡x4sin⁡2yHence, by implicit differentiation we have dydx=cos⁡x4sin⁡2y.

Jeffrey Jordon · Accepted Answer

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Find dy / dx by implicit differentiation. \sin x+2\cos 2y=1

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