Questions and Answers
In implicit differentiation, when finding the derivative of $x^2 + y^2 = 9$ with respect to $x$, what is the correct derivative of $y$ with respect to $x$?
$-x/y$
For the equation $rac{d}{dx}(rac{3x^2 - y\ ext{cos}(xy)}{ ext{sin}(xy)})$, what is the result when simplifying?
$0$
When finding the derivative of $ ext{sin}(xy)$ using implicit differentiation, what is the correct term that appears in the expression?
$y\text{cos}(xy)$
What rule is crucial to apply when differentiating $ ext{sin}(xy)$ implicitly in the provided examples?
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In implicit differentiation, what can make the process challenging?
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Study Notes
Implicit Differentiation: A Deep Dive into Derivative Rules and Examples
Implicit differentiation is a fundamental concept in calculus that allows us to find the derivative of a function whose equation is given implicitly—in other words, when the relationship between variables is not explicitly stated in the form of a function like (y = f(x)). Think of it as a superpower that helps us leverage our existing derivative rules for finding the derivative of an implicit function.
Implicit Differentiation Rules
To find the derivative of an implicit function, we apply the same rules of differentiation that we use for explicit functions, but we do so with respect to both variables, (x) and (y).
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Differentiation rules: Apply the regular derivative rules for each term in the equation. For example, if we have (xy^2 + x^2 = 5), then (\frac{d}{dx}(xy^2) = \frac{d}{dx}(x^2)).
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Chain rule: Apply the chain rule whenever necessary. For example, in (\sin(xy) = x^3), we must differentiate both (\sin(xy)) and (xy).
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Implicit differentiation: Treat (y) as a function of (x) and differentiate both sides of the equation with respect to (x).
Examples of Implicit Differentiation
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(xy = 3x^2 + 4) [ \frac{d}{dx}(xy) = \frac{d}{dx}(3x^2 + 4) ] Using the product rule, we get: [ y + x\frac{dy}{dx} = 6x ] Solving for (\frac{dy}{dx}), we get: [ \frac{dy}{dx} = \frac{6x - y}{x} ]
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(x^2 + y^2 = 9) Differentiation with respect to (x): [ 2x + 2y\frac{dy}{dx} = 0 ] Solving for (\frac{dy}{dx}), we get: [ \frac{dy}{dx} = -\frac{x}{y} ]
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(\sin(xy) = x^3) [ \frac{d}{dx}(\sin(xy)) = \frac{d}{dx}(x^3) ] Using the chain rule, we get: [ y\cos(xy) + x\frac{dy}{dx}\sin(xy) = 3x^2 ] Solving for (\frac{dy}{dx}), we get: [ \frac{dy}{dx} = \frac{3x^2 - y\cos(xy)}{\sin(xy)} ]
Final Thoughts
Implicit differentiation is a powerful tool, but it can also be a bit tricky because you must be careful to apply the chain rule and other derivative rules correctly. By working through several practice examples, you'll build an intuition for how to differentiate implicitly, allowing you to tackle more complex problems in your calculus studies.
[Remark: Since the search results do not directly provide information about implicit differentiation, the content in this article is based on my own knowledge and understanding of the subject.]
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Description
Test your understanding of implicit differentiation with this quiz that covers derivative rules and examples for finding the derivative of functions given implicitly. Explore differentiation rules, the chain rule, and practice solving implicit differentiation problems to strengthen your calculus skills.
