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

Problem

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

Solution

1. Apply logarithm properties to simplify the expression before differentiating. Use the rule ln(a/b)=ln(a)−ln(b)

ln(x/(1+x2))=ln(x)−ln(1+x2)

2. Differentiate the first term using the rule d(ln(x))/d(x)=1/x

d(ln(x))/d(x)=1/x

3. Differentiate the second term using the chain rule d(ln(u))/d(x)=1/u⋅d(u)/d(x) where u=1+x2

d(ln(1+x2))/d(x)=1/(1+x2)⋅2*x

4. Combine the results to find the derivative of the entire expression.

d(y)/d(x)=1/x−(2*x)/(1+x2)

5. Find a common denominator to simplify the final expression. The common denominator is x*(1+x2)

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

6. Subtract the numerators and simplify.

d(y)/d(x)=(1−x2)/(x*(1+x2))

Final Answer

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

Want more problems? Check here!