If y = log ≤ft(1−x²/1+x²), then dy/dx is equal to

Correct Answer is : −4x1−x4 −4 x/1−x^41−x4−4x

Correct Answer is : −4x1−x4 −4 x/1−x^41−x4−4x

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AP DSC School Assistant Math Model Test 3

Section: Subject Specific

Question : 115 of 160

Marks: +1, -0

If y = log

\rm \left(\frac{1−x^2}{1+x^2}\right)

\rm \frac{dy}{dx}

is equal to

Solution:

Concept:

Use the logarithm property to express the fraction as a difference of logs, then apply the derivative of

\log x

and the chain rule.

Explanation:

Given

y = \log\left(\frac{1-x^2}{1+x^2}\right)

Using

\log(a/b) = \log a - \log b

, we get

y = \log(1-x^2) - \log(1+x^2)

Differentiate term by term:

\frac{d}{dx}\log(1-x^2) = \frac{1}{1-x^2} \cdot (-2x)

and

\frac{d}{dx}\log(1+x^2) = \frac{1}{1+x^2} \cdot (2x)

\frac{dy}{dx} = \frac{-2x}{1-x^2} - \frac{2x}{1+x^2} = -2x\left(\frac{1}{1-x^2} + \frac{1}{1+x^2}\right)

Combine the fractions:

\frac{1}{1-x^2} + \frac{1}{1+x^2} = \frac{(1+x^2)+(1-x^2)}{(1-x^2)(1+x^2)} = \frac{2}{1-x^4}

Thus

\frac{dy}{dx} = -2x \cdot \frac{2}{1-x^4} = \frac{-4x}{1-x^4}

Answer:

\frac{-4x}{1-x^4}

which corresponds to option B.

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