Find the Derivative - d/dx arctan(x)

Problem

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

Solution

1. Set up the equation by letting y=arctan(x) which implies x=tan(y)

2. Differentiate both sides with respect to x using implicit differentiation.

d(x)/d(x)=d(tan(y))/d(x)

3. Apply the chain rule to the right side of the equation.

1=sec2(y)d(y)/d(x)

4. Solve for the derivative d(y)/d(x) by isolating it.

d(y)/d(x)=1/sec2(y)

5. Use the trigonometric identity sec2(y)=1+tan2(y) to rewrite the expression.

d(y)/d(x)=1/(1+tan2(y))

6. Substitute x back into the expression using the original relationship x=tan(y)

d(y)/d(x)=1/(1+x2)

Final Answer

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

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