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

Problem

d()/d(x)*(x+1)2*(x2+1)(−3)

Solution

1. Identify the rule needed for the derivative, which is the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=(x+1)2 and v=(x2+1)(−3)

2. Differentiate the first part u=(x+1)2 using the power rule and chain rule.

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

3. Differentiate the second part v=(x2+1)(−3) using the power rule and chain rule.

d(x2+1)/d(x)=−3*(x2+1)(−4)*(2*x)

d(x2+1)/d(x)=−6*x*(x2+1)(−4)

4. Apply the product rule by combining the derivatives.

d(y)/d(x)=(x+1)2*(−6*x*(x2+1)(−4))+(x2+1)(−3)*(2*(x+1))

5. Factor out the greatest common factors, which are 2*(x+1) and (x2+1)(−4)

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

6. Simplify the expression inside the brackets.

−3*x2−3*x+x2+1=−2*x2−3*x+1

7. Combine all terms into a single rational expression.

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

Final Answer

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

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