If `log_(10)((x^(3) - y^(3))/ (x^(3)+y^(3))) = 2`, then `(dy)/(dx)=`

Options

A. `x/y`
B. `-y/x`
C. `-x/y`
D. `y/x`

Answer

To solve the problem step by step, we start with the given equation:

\[ \log_{10}\left(\frac{x^3 - y^3}{x^3 + y^3}\right) = 2 \]

### Step 1: Convert the logarithmic equation to exponential form
Using the properties of logarithms, we can rewrite the equation as:

\[ \frac{x^3 - y^3}{x^3 + y^3} = 10^2 \]

This simplifies to:

\[ \frac{x^3 - y^3}{x^3 + y^3} = 100 \]

**Hint:** Remember that if \(\log_b(a) = c\), then \(a = b^c\).

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### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ x^3 - y^3 = 100(x^3 + y^3) \]

Expanding the right side:

\[ x^3 - y^3 = 100x^3 + 100y^3 \]

**Hint:** When cross-multiplying, ensure you distribute correctly on both sides.

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### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate terms involving \(y^3\):

\[ x^3 - 100x^3 - 100y^3 = y^3 \]

This simplifies to:

\[ -99x^3 = 101y^3 \]

**Hint:** Keep track of the signs when moving terms from one side of the equation to the other.

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### Step 4: Express \(y^3\) in terms of \(x^3\)
We can now express \(y^3\) in terms of \(x^3\):

\[ y^3 = -\frac{99}{101}x^3 \]

**Hint:** Make sure to simplify fractions correctly.

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### Step 5: Take the cube root to find \(y\)
Taking the cube root of both sides gives:

\[ y = \left(-\frac{99}{101}\right)^{1/3} x \]

**Hint:** Remember that the cube root of a negative number is also negative.

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### Step 6: Differentiate both sides with respect to \(x\)
Now, we differentiate both sides with respect to \(x\):

\[ \frac{dy}{dx} = \left(-\frac{99}{101}\right)^{1/3} \]

**Hint:** When differentiating, use the constant multiple rule.

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### Final Result
Thus, we find:

\[ \frac{dy}{dx} = \left(-\frac{99}{101}\right)^{1/3} \]

The correct answer corresponds to the fourth option.

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