Question

Use the method of implicit differentiation to determine the derivative of the following functions:

1. xsiny +ysinx=1

2. y + xcosy=x^2 y

Answer 1

1.

Differentiate both sides with respect to $x$

$$\dfrac{d}{dx}(x\sin y+y\sin x)=\dfrac{d}{dx}(1)$$

Use the Chain Rule

$$\sin y+x\cos y\cdot\dfrac{dy}{dx}+\sin x\cdot\dfrac{dy}{dx}+y\cos x=0$$

Solve for $\dfrac{dy}{dx}$

$$\dfrac{dy}{dx}=-\dfrac{\sin y+y\cos x}{x\cos y+\sin x}$$

2.

Differentiate both sides with respect to $x$

$$\dfrac{d}{dx}(y+x\cos y)=\dfrac{d}{dx}(x^2y)$$

Use the Chain Rule

$$\dfrac{dy}{dx}+\cos y-x\sin y\cdot\dfrac{dy}{dx}=2xy+x^2\cdot\dfrac{dy}{dx}$$

Solve for $\dfrac{dy}{dx}$

$$\dfrac{dy}{dx}=\dfrac{\cos y-2xy}{x^2+x\sin y-1}$$