Find the Derivative - d/dx arctan(4x)

Problem

d(arctan(4*x))/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=4*x

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=4*x with respect to x which gives d(u)/d(x)=4

4. Combine the results using the Chain Rule: d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

5. Substitute 4*x back into the expression for u and simplify.

Final Answer

d(arctan(4*x))/d(x)=4/(1+16*x2)

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