Find the Derivative - d/dx y=arctan(x/2)

Problem

d()/d(x)*arctan(x/2)

Solution

1. Identify the outer function as arctan(u) and the inner function as u=x/2

2. Apply the chain rule which states that d(arctan(u))/d(x)=1/(1+u2)⋅d(u)/d(x)

3. Differentiate the inner function u=x/2 to get d(u)/d(x)=1/2

4. Substitute the expressions for u and d(u)/d(x) into the chain rule formula.

d(arctan(x/2))/d(x)=1/(1+(x/2)2)⋅1/2

5. Simplify the denominator by squaring the fraction.

d(arctan(x/2))/d(x)=1/(1+(x2)/4)⋅1/2

6. Combine the terms by multiplying the fractions.

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

7. Distribute the constant in the denominator to reach the final form.

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

8. Multiply the numerator and denominator by 2 to eliminate the fraction within the denominator.

d(arctan(x/2))/d(x)=2/(4+x2)

Final Answer

d(arctan(x/2))/d(x)=2/(x2+4)

Want more problems? Check here!