1 ?/4 1. evaluate tanjy dydx
LectureNotes said 1 ?/4 1. evaluate tanjy dydx
Answer:
It seems like the question may contain some formatting issues or typos. However, I’ll do my best to interpret and provide a detailed response. The question appears to involve evaluating the derivative of a function involving trigonometric notation, specifically \tan(y), with respect to x.
Let’s break down what we need to do:
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Understand the Function:
- Assume y is a function of x, i.e., y = f(x).
- Evaluate the derivative \frac{d}{dx}(\tan(y)).
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Apply the Chain Rule:
- Because y is a function of x, we will use the chain rule for differentiation.
Chain Rule Application
If we have a composite function u(v(x)), the chain rule states that:
\frac{d}{dx}u(v(x)) = \frac{du}{dv} \cdot \frac{dv}{dx}
For our specific function, u = \tan(y) and v = y. Therefore,
\frac{d}{dx} (\tan(y)) = \frac{d}{dy} (\tan(y)) \cdot \frac{dy}{dx}
Derivative Computation
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Derivative of \tan(y) with Respect to y:
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The derivative of \tan(y) with respect to y is \sec^2(y).
\frac{d}{dy} (\tan(y)) = \sec^2(y)
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Express \frac{dy}{dx}:
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This is simply the notation for the derivative of y with respect to x.
\frac{dy}{dx} = \frac{d}{dx} y
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Final Result
Putting it all together using the chain rule:
\frac{d}{dx} (\tan(y)) = \sec^2(y) \cdot \frac{dy}{dx}
Final Answer:
\frac{d}{dx} (\tan(y)) = \sec^2(y) \cdot \frac{dy}{dx}
This is the derivative of \tan(y) with respect to x. If you have specific values or conditions for y, these can be substituted into the formula for a more concrete answer.