How Can I Reconcile The Apparent Discrepancy In The Derivative Of A Function When Using Implicit Differentiation Versus Using The Chain Rule And The Quotient Rule, Specifically When Dealing With Trigonometric Functions That Have Multiple Nested Layers Of Differentiation, Such As The Derivative Of Arctan(sin(x)/cos(x))?

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To reconcile the apparent discrepancy when differentiating the function arctan(sin(x)/cos(x)) using different methods, follow these steps:

  1. Recognize the Function Structure: The function is arctan of the quotient sin(x)/cos(x), which simplifies to arctan(tan(x)).

  2. Simplify the Function: Since arctan(tan(x)) = x (within the principal value range), the derivative is straightforward.

  3. Differentiate Using Chain Rule and Quotient Rule:

    • Let y = arctan(u), where u = sin(x)/cos(x).
    • dy/dx = (1 / (1 + u²)) * du/dx.
    • Compute du/dx using the quotient rule: [cos(x)cos(x) + sin(x)sin(x)] / cos²(x) = 1 / cos²(x).
    • Substitute back: dy/dx = (1 / (1 + tan²(x))) * (1 / cos²(x)) = 1.

  4. Conclusion: Both methods yield dy/dx = 1, confirming consistency.

Final Answer The derivative is \boxed{1}.