Find dydx by implicit differentiation. sin(x) + cos(y) = 7x − 2y
The correct answer and explanation is:
To find dydx\frac{dy}{dx} by implicit differentiation for the equation: sin(x)+cos(y)=7x−2y\sin(x) + \cos(y) = 7x – 2y
Step-by-Step Solution:
Differentiate both sides of the equation implicitly with respect to xx. Remember:
- yy is a function of xx, so when differentiating a term involving yy, apply the chain rule (i.e., multiply by dydx\frac{dy}{dx}).
Differentiate the left side: ddx[sin(x)]+ddx[cos(y)]=cos(x)−sin(y)⋅dydx\frac{d}{dx}[\sin(x)] + \frac{d}{dx}[\cos(y)] = \cos(x) – \sin(y) \cdot \frac{dy}{dx}
Differentiate the right side: ddx[7x]−ddx[2y]=7−2⋅dydx\frac{d}{dx}[7x] – \frac{d}{dx}[2y] = 7 – 2 \cdot \frac{dy}{dx}
Now, set the derivatives equal: cos(x)−sin(y)⋅dydx=7−2⋅dydx\cos(x) – \sin(y) \cdot \frac{dy}{dx} = 7 – 2 \cdot \frac{dy}{dx}
Now, solve for dydx\frac{dy}{dx}:
- Move all terms with dydx\frac{dy}{dx} to one side:
−sin(y)⋅dydx+2⋅dydx=7−cos(x)- \sin(y) \cdot \frac{dy}{dx} + 2 \cdot \frac{dy}{dx} = 7 – \cos(x)
- Factor out dydx\frac{dy}{dx}:
(2−sin(y))⋅dydx=7−cos(x)(2 – \sin(y)) \cdot \frac{dy}{dx} = 7 – \cos(x)
- Solve for dydx\frac{dy}{dx}:
dydx=7−cos(x)2−sin(y)\frac{dy}{dx} = \frac{7 – \cos(x)}{2 – \sin(y)}
Final Answer:
dydx=7−cos(x)2−sin(y)\boxed{\frac{dy}{dx} = \frac{7 – \cos(x)}{2 – \sin(y)}}
Explanation (300 words):
Implicit differentiation is a powerful tool used when dealing with equations that define yy implicitly as a function of xx, rather than explicitly (i.e., not in the form y=f(x)y = f(x)). In this case, the equation involves both xx and yy intermixed within trigonometric functions and on both sides of the equation: sin(x)+cos(y)=7x−2y\sin(x) + \cos(y) = 7x – 2y
Since we are interested in how yy changes with respect to xx, we apply implicit differentiation. The key idea is that when differentiating a term involving yy, we treat yy as a function of xx, so we must apply the chain rule. For example, the derivative of cos(y)\cos(y) with respect to xx is −sin(y)⋅dydx-\sin(y) \cdot \frac{dy}{dx}, because we differentiate the outer function (cosine) and multiply by the derivative of the inner function yy with respect to xx.
After differentiating both sides, we collect all terms involving dydx\frac{dy}{dx} on one side of the equation and factor it out. Finally, solving algebraically yields the derivative.
The final expression: dydx=7−cos(x)2−sin(y)\frac{dy}{dx} = \frac{7 – \cos(x)}{2 – \sin(y)}
gives the rate of change of yy with respect to xx in terms of both xx and yy. This is often the result when using implicit differentiation—because yy is defined implicitly, the derivative is also in terms of both variables.