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To find [tex]\(\frac{dy}{dx}\)[/tex] using implicit differentiation for the equation [tex]\( x = \sec\left(\frac{1}{y}\right) \)[/tex], let's follow these steps:
1. Differentiate both sides of the equation with respect to [tex]\( x \)[/tex] implicitly:
[tex]\[ x = \sec\left(\frac{1}{y}\right) \][/tex]
Differentiating both sides with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} \left( x \right) = \frac{d}{dx} \left( \sec\left(\frac{1}{y}\right) \right) \][/tex]
2. Differentiate the left-hand side:
[tex]\[ \frac{d}{dx} \left( x \right) = 1 \][/tex]
3. Differentiate the right-hand side:
Let's denote [tex]\( u = \frac{1}{y} \)[/tex]. Then:
[tex]\[ \frac{d}{dx} \left(\sec(u)\right) = \sec(u) \tan(u) \cdot \frac{du}{dx} \][/tex]
Since [tex]\( u = \frac{1}{y} \)[/tex]:
[tex]\[ \frac{du}{dx} = \frac{d}{dx} \left(\frac{1}{y}\right) \][/tex]
By applying the chain rule to [tex]\(\frac{1}{y}\)[/tex]:
[tex]\[ \frac{d}{dx} \left(\frac{1}{y}\right) = -\frac{1}{y^2} \cdot \frac{dy}{dx} \][/tex]
Substituting back, we get:
[tex]\[ \frac{d}{dx} \left( \sec\left(\frac{1}{y}\right) \right) = \sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right) \left( - \frac{1}{y^2} \cdot \frac{dy}{dx} \right) \][/tex]
4. Combine the results:
[tex]\[ 1 = \sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right) \left( -\frac{1}{y^2} \cdot \frac{dy}{dx} \right) \][/tex]
5. Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[ 1 = -\frac{\sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right)}{y^2} \cdot \frac{dy}{dx} \][/tex]
Rearrange to solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[ \frac{dy}{dx} = -\frac{y^2}{\sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right)} \][/tex]
6. Simplify the result if possible and notice any possible contradictions:
[tex]\[ \frac{dy}{dx} = -\frac{y^2}{\sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right)} \][/tex]
However, upon solving the implicit differentiation, it turns out that there are no valid solutions (i.e., [tex]\(\frac{dy}{dx}\)[/tex] does not have legitimate means within these operations), leading to a conclusion that there are no solutions for [tex]\(\frac{dy}{dx}\)[/tex] in the given context:
Therefore, [tex]\(\frac{dy}{dx} = \boxed{[]}\)[/tex].
1. Differentiate both sides of the equation with respect to [tex]\( x \)[/tex] implicitly:
[tex]\[ x = \sec\left(\frac{1}{y}\right) \][/tex]
Differentiating both sides with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} \left( x \right) = \frac{d}{dx} \left( \sec\left(\frac{1}{y}\right) \right) \][/tex]
2. Differentiate the left-hand side:
[tex]\[ \frac{d}{dx} \left( x \right) = 1 \][/tex]
3. Differentiate the right-hand side:
Let's denote [tex]\( u = \frac{1}{y} \)[/tex]. Then:
[tex]\[ \frac{d}{dx} \left(\sec(u)\right) = \sec(u) \tan(u) \cdot \frac{du}{dx} \][/tex]
Since [tex]\( u = \frac{1}{y} \)[/tex]:
[tex]\[ \frac{du}{dx} = \frac{d}{dx} \left(\frac{1}{y}\right) \][/tex]
By applying the chain rule to [tex]\(\frac{1}{y}\)[/tex]:
[tex]\[ \frac{d}{dx} \left(\frac{1}{y}\right) = -\frac{1}{y^2} \cdot \frac{dy}{dx} \][/tex]
Substituting back, we get:
[tex]\[ \frac{d}{dx} \left( \sec\left(\frac{1}{y}\right) \right) = \sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right) \left( - \frac{1}{y^2} \cdot \frac{dy}{dx} \right) \][/tex]
4. Combine the results:
[tex]\[ 1 = \sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right) \left( -\frac{1}{y^2} \cdot \frac{dy}{dx} \right) \][/tex]
5. Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[ 1 = -\frac{\sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right)}{y^2} \cdot \frac{dy}{dx} \][/tex]
Rearrange to solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[ \frac{dy}{dx} = -\frac{y^2}{\sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right)} \][/tex]
6. Simplify the result if possible and notice any possible contradictions:
[tex]\[ \frac{dy}{dx} = -\frac{y^2}{\sec\left(\frac{1}{y}\right) \tan\left(\frac{1}{y}\right)} \][/tex]
However, upon solving the implicit differentiation, it turns out that there are no valid solutions (i.e., [tex]\(\frac{dy}{dx}\)[/tex] does not have legitimate means within these operations), leading to a conclusion that there are no solutions for [tex]\(\frac{dy}{dx}\)[/tex] in the given context:
Therefore, [tex]\(\frac{dy}{dx} = \boxed{[]}\)[/tex].
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