Find the Derivative - d/dx (-4x)/((x^2-1)^2)

Problem

d()/d(x)(−4*x)/((x2−1)2)

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction, use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions. Let u=−4*x and v=(x2−1)2

3. Differentiate the numerator.

(d(−)*4*x)/d(x)=−4

4. Differentiate the denominator using the chain rule.

d(x2−1)/d(x)=2*(x2−1)⋅2*x

d(x2−1)/d(x)=4*x*(x2−1)

5. Apply the quotient rule formula.

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

6. Factor out the common term (x2−1) from the numerator to simplify.

d()/d(x)(−4*x)/((x2−1)2)=((x2−1)*[−4*(x2−1)+16*x2])/((x2−1)4)

7. Cancel the common factor (x2−1) and simplify the expression inside the brackets.

d()/d(x)(−4*x)/((x2−1)2)=(−4*x2+4+16*x2)/((x2−1)3)

8. Combine like terms in the numerator.

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

Final Answer

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

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