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Chapter 2: Problem 17

Find \(d y / d x\) by implicit differentiation. $$ y=\sin (x y) $$

Short Answer

Expert verified
\[ \frac{dy}{dx} = \frac{y\cos(xy)}{1 - x\cos(xy)} \]

Step by step solution

01

Differentiate Both Sides w.r.t. x

Differentiate the given function \(y = \sin(xy)\) with respect to \(x\). By applying the chain rule, the derivative of the outer function (sine) times the derivative of the inner function. The derivative of \(y(x)\) with respect to \(x\) is \(dy/dx\). \[ \frac{dy}{dx} = \cos(xy) * \frac{d}{dx}(xy) \]
02

Apply the Product rule

Now to further simplify the right side, apply the product rule on \(\frac{d}{dx}(xy)\) which states that the derivative of gcd two functions is the derivative of the first times the second, plus the first times the derivative of the second function. In this case; \[ \frac{d}{dx}(xy)=x\frac{dy}{dx}+y \]
03

Substitute Back

Substitute \(\frac{d}{dx}(xy)=x\frac{dy}{dx}+y\) back into the equation derived from Step 1. \[ \frac{dy}{dx}=\cos(xy) * (x\frac{dy}{dx}+y) \]
04

Solving for \(dy/dx\)

The aim is to solve for \(dy/dx\). So, shift the term containing \(dy/dx\) on one side and rest everything on the other side of the equation and then factor it out to solve for \(dy/dx\). \( \frac{dy}{dx} - x\cos(xy)\frac{dy}{dx} = y\cos(xy) \) Factor out \( dy/dx \) on the left to get: \( \frac{dy}{dx} [1 - x\cos(xy)] = y\cos(xy) \) Finally, solve for \( dy/dx \): \( \frac{dy}{dx} = \frac{y\cos(xy)}{1 - x\cos(xy)} \)

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