Find the Derivative - d/dx y=((x^2+1)/(x^2-1))^3

Problem

d()/d(x)*((x2+1)/(x2−1))3

Solution

1. Apply the power rule and the chain rule by treating the inner fraction as u where y=u3

d(y)/d(x)=3*((x2+1)/(x2−1))2⋅d()/d(x)(x2+1)/(x2−1)

2. Apply the quotient rule to differentiate the inner expression (x2+1)/(x2−1)

d()/d(x)(x2+1)/(x2−1)=((x2−1)⋅d(x2+1)/d(x)−(x2+1)⋅d(x2−1)/d(x))/((x2−1)2)

3. Differentiate the terms inside the quotient rule formula.

d()/d(x)(x2+1)/(x2−1)=((x2−1)*(2*x)−(x2+1)*(2*x))/((x2−1)2)

4. Simplify the numerator by distributing and combining like terms.

d()/d(x)(x2+1)/(x2−1)=(2*x3−2*x−2*x3−2*x)/((x2−1)2)

d()/d(x)(x2+1)/(x2−1)=(−4*x)/((x2−1)2)

5. Substitute the result back into the chain rule expression.

d(y)/d(x)=3*((x2+1)/(x2−1))2⋅(−4*x)/((x2−1)2)

6. Combine the factors to reach the final simplified form.

d(y)/d(x)=(−12*x*(x2+1)2)/((x2−1)2*(x2−1)2)

d(y)/d(x)=(−12*x*(x2+1)2)/((x2−1)4)

Final Answer

d()/d(x)*((x2+1)/(x2−1))3=(−12*x*(x2+1)2)/((x2−1)4)

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