Find the Derivative - d/dx (5x)/(1+x^2)

Problem

d()/d(x)(5*x)/(1+x2)

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction of two functions, 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=5*x and v=1+x2

3. Differentiate the individual components. Find (d(5)*x)/d(x)=5 and d(1+x2)/d(x)=2*x

4. Substitute these values into the quotient rule formula.

d()/d(x)(5*x)/(1+x2)=((1+x2)*(5)−(5*x)*(2*x))/((1+x2)2)

5. Expand the terms in the numerator.

d()/d(x)(5*x)/(1+x2)=(5+5*x2−10*x2)/((1+x2)2)

6. Combine like terms in the numerator to reach the final simplified form.

d()/d(x)(5*x)/(1+x2)=(5−5*x2)/((1+x2)2)

Final Answer

d()/d(x)(5*x)/(1+x2)=(5*(1−x2))/((1+x2)2)

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