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

Problem

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

Solution

1. Identify the quotient rule, which states that for a function y=u/v the derivative is (vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions:

u=4*x

v=(x2+1)2

3. Differentiate u with respect to x

d(u)/d(x)=4

4. Differentiate v using the chain rule:

d(v)/d(x)=2*(x2+1)⋅2*x

d(v)/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:

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

7. Simplify the expression by canceling one (x2+1) term and distributing the constants:

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)=(4−12*x2)/((x2+1)3)

Final Answer

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

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