Find the Derivative - d/dx arctan(x/3)

Problem

d(arctan(x/3))/d(x)

Solution

1. Identify the outer function and the inner function to apply the chain rule. The outer function is arctan(u) and the inner function is u=x/3

2. Apply the formula for the derivative of the arctangent function, which is d(arctan(u))/d(u)=1/(1+u2)

3. Differentiate the inner function u=1/3*x with respect to x which yields d(u)/d(x)=1/3

4. Combine the results using the chain rule: d(arctan(u))/d(x)=d(arctan(u))/d(u)⋅d(u)/d(x)

d(arctan(x/3))/d(x)=1/(1+(x/3)2)⋅1/3

5. Simplify the expression by squaring the fraction in the denominator.

d(arctan(x/3))/d(x)=1/(1+(x2)/9)⋅1/3

6. Distribute the 1/3 into the denominator of the first fraction.

d(arctan(x/3))/d(x)=1/(3*(1+(x2)/9))

7. Finalize the simplification by multiplying the terms in the denominator.

d(arctan(x/3))/d(x)=1/(3+(x2)/3)

8. Rewrite the expression in a standard form by multiplying the numerator and denominator by 3

d(arctan(x/3))/d(x)=3/(9+x2)

Final Answer

d(arctan(x/3))/d(x)=3/(9+x2)

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