Math Problem Statement
determina y'
Solution
To determine from the given equation:
we will follow these steps:
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Implicit Differentiation: Since is a function of , we differentiate both sides of the equation with respect to , treating as a function of .
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Apply Chain Rule: Wherever appears, apply the chain rule.
Let’s begin the differentiation.
Step 1: Differentiate both sides with respect to :
Differentiating the left side term-by-term:
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First term:
Using the product rule:
Now, for the derivative of :
[ \frac{d}{dx}\left[\log\left(\frac{x}{y}\right)\right] = \frac{1}{\frac{x}{y}} \cdot \frac{d}{dx}\left[\frac{x}{y}\right] ]
Simplifying :
Therefore:
So, the derivative of the first term is:
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Second term:
Apply the chain rule:
The derivative of the left side:
The derivative of the right side is 0.
Step 2: Combine and solve for :
Set the derivatives equal:
Factor out :
[ y'\left[ -\ln(10) + x\sin(xy)\right] = y\left[\sin(xy) - \frac{\ln(10)}{x}\right] ]
Finally, solve for :
This is the derivative for the given equation.
Do you want more details or have any questions? Here are some related questions to explore:
- How would the derivative change if the argument of the logarithm was different?
- What would happen if we used a different logarithmic base?
- How does implicit differentiation work in cases where both and are in trigonometric functions?
- How can we simplify expressions involving logarithmic and trigonometric functions before differentiation?
- How would you interpret the result geometrically?
Tip: Always make sure to apply the chain rule correctly when differentiating composite functions involving products of and .
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Chain Rule
Logarithmic Functions
Trigonometric Functions
Formulas
Chain rule for implicit differentiation
Theorems
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Suitable Grade Level
Advanced College