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

Problem

d()/d(x)(1−x2)/(1+x2)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two expressions, u=1−x2 and v=1+x2 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Differentiate the numerator u=1−x2 with respect to x

d(1−x2)/d(x)=−2*x

3. Differentiate the denominator v=1+x2 with respect to x

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

4. Substitute these components into the quotient rule formula.

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

5. Expand the terms in the numerator.

(−2*x−2*x3−(2*x−2*x3))/((1+x2)2)

6. Simplify the numerator by combining like terms.

(−2*x−2*x3−2*x+2*x3)/((1+x2)2)

(−4*x)/((1+x2)2)

Final Answer

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

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