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

Problem

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

Solution

1. Identify the function as the inverse tangent function, ƒ(x)=arctan(x) which is also written as tan(x)(−1)

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

3. Differentiate implicitly both sides with respect to x to find d(y)/d(x)

4. Apply the chain rule to the left side, resulting in sec2(y)d(y)/d(x)=1

5. Solve for the derivative by dividing both sides by sec2(y) giving d(y)/d(x)=1/sec2(y)

6. Use the trigonometric identity sec2(y)=1+tan2(y) to rewrite the expression in terms of y

7. Substitute the original relationship tan(y)=x back into the expression to get d(y)/d(x)=1/(1+x2)

Final Answer

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

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