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

Problem

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

Solution

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

2. Apply the chain rule, which states that d(ƒ(u))/d(x)=d(ƒ)/d(u)⋅d(u)/d(x)

3. Differentiate the outer function using the rule d(arctan(u))/d(u)=1/(1+u2)

4. Differentiate the inner function d()/d(x)x/2=1/2

5. Substitute the expressions back into the chain rule formula.

d(arctan(x/2))/d(x)=1/(1+(x/2)2)⋅1/2

6. Simplify the denominator by squaring the fraction.

d(arctan(x/2))/d(x)=1/(1+(x2)/4)⋅1/2

7. Combine the terms by multiplying the fractions.

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

8. Distribute the constant to reach the final simplified form.

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

9. Multiply the numerator and denominator by 2 to clear the fraction in the denominator.

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

Final Answer

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

Want more problems? Check here!