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

Problem

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

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is ƒ(u)=arctan(u) and the inner function is u=cos(x)

2. Recall the derivative formula for the inverse tangent function, which is d(arctan(u))/d(u)=1/(1+u2)

3. Apply the Chain Rule by multiplying the derivative of the outer function by the derivative of the inner function.

d(arctan(cos(x)))/d(x)=1/(1+(cos(x))2)⋅d(cos(x))/d(x)

4. Differentiate the inner function cos(x) which results in −sin(x)

d(arctan(cos(x)))/d(x)=1/(1+cos2(x))⋅(−sin(x))

5. Simplify the expression by multiplying the terms together.

d(arctan(cos(x)))/d(x)=(−sin(x))/(1+cos2(x))

Final Answer

d(arctan(cos(x)))/d(x)=−sin(x)/(1+cos2(x))

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