Find dydx by implicit differentiation. sin(x) + cos(y) = 7x \u2212 2y<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n To find dydx\\frac{dy}{dx} by implicit differentiation<\/strong> for the equation: sin\u2061(x)+cos\u2061(y)=7x\u22122y\\sin(x) + \\cos(y) = 7x – 2y<\/p>\n\n\n\n Differentiate both sides of the equation implicitly<\/strong> with respect to xx. Remember:<\/p>\n\n\n\n Differentiate the left side:<\/strong> ddx[sin\u2061(x)]+ddx[cos\u2061(y)]=cos\u2061(x)\u2212sin\u2061(y)\u22c5dydx\\frac{d}{dx}[\\sin(x)] + \\frac{d}{dx}[\\cos(y)] = \\cos(x) – \\sin(y) \\cdot \\frac{dy}{dx}<\/p>\n\n\n\n Differentiate the right side:<\/strong> ddx[7x]\u2212ddx[2y]=7\u22122\u22c5dydx\\frac{d}{dx}[7x] – \\frac{d}{dx}[2y] = 7 – 2 \\cdot \\frac{dy}{dx}<\/p>\n\n\n\n Now, set the derivatives equal: cos\u2061(x)\u2212sin\u2061(y)\u22c5dydx=7\u22122\u22c5dydx\\cos(x) – \\sin(y) \\cdot \\frac{dy}{dx} = 7 – 2 \\cdot \\frac{dy}{dx}<\/p>\n\n\n\n Now, solve for dydx\\frac{dy}{dx}:<\/p>\n\n\n\n \u2212sin\u2061(y)\u22c5dydx+2\u22c5dydx=7\u2212cos\u2061(x)- \\sin(y) \\cdot \\frac{dy}{dx} + 2 \\cdot \\frac{dy}{dx} = 7 – \\cos(x)<\/p>\n\n\n\n (2\u2212sin\u2061(y))\u22c5dydx=7\u2212cos\u2061(x)(2 – \\sin(y)) \\cdot \\frac{dy}{dx} = 7 – \\cos(x)<\/p>\n\n\n\n dydx=7\u2212cos\u2061(x)2\u2212sin\u2061(y)\\frac{dy}{dx} = \\frac{7 – \\cos(x)}{2 – \\sin(y)}<\/p>\n\n\n\n dydx=7\u2212cos\u2061(x)2\u2212sin\u2061(y)\\boxed{\\frac{dy}{dx} = \\frac{7 – \\cos(x)}{2 – \\sin(y)}}<\/p>\n\n\n\n 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\u2061(x)+cos\u2061(y)=7x\u22122y\\sin(x) + \\cos(y) = 7x – 2y<\/p>\n\n\n\n Since we are interested in how yy changes with respect to xx, we apply implicit differentiation<\/strong>. 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<\/strong>. For example, the derivative of cos\u2061(y)\\cos(y) with respect to xx is \u2212sin\u2061(y)\u22c5dydx-\\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.<\/p>\n\n\n\n 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.<\/p>\n\n\n\n The final expression: dydx=7\u2212cos\u2061(x)2\u2212sin\u2061(y)\\frac{dy}{dx} = \\frac{7 – \\cos(x)}{2 – \\sin(y)}<\/p>\n\n\n\n gives the rate of change of yy with respect to xx<\/strong> in terms of both xx and yy. This is often the result when using implicit differentiation\u2014because yy is defined implicitly, the derivative is also in terms of both variables.<\/p>\n","protected":false},"excerpt":{"rendered":" Find dydx by implicit differentiation. sin(x) + cos(y) = 7x \u2212 2y The correct answer and explanation is: To find dydx\\frac{dy}{dx} by implicit differentiation for the equation: sin\u2061(x)+cos\u2061(y)=7x\u22122y\\sin(x) + \\cos(y) = 7x – 2y Step-by-Step Solution: Differentiate both sides of the equation implicitly with respect to xx. Remember: Differentiate the left side: ddx[sin\u2061(x)]+ddx[cos\u2061(y)]=cos\u2061(x)\u2212sin\u2061(y)\u22c5dydx\\frac{d}{dx}[\\sin(x)] + \\frac{d}{dx}[\\cos(y)] […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-17734","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/17734","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=17734"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/17734\/revisions"}],"predecessor-version":[{"id":17735,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/17734\/revisions\/17735"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=17734"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=17734"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=17734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step-by-Step Solution:<\/h3>\n\n\n\n
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\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
\n\n\n\nExplanation (300 words):<\/h3>\n\n\n\n