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Differentiate f(x)=1−cos⁡(4x)1+cos⁡(4x)f(x)=\sqrt{\frac{1-\cos(4x)}{1+\cos(4x)}}

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Q. Differentiate f(x)=1−cos⁡(4x)1+cos⁡(4x)f(x)=\sqrt{\frac{1-\cos(4x)}{1+\cos(4x)}}
  1. Identify Function: Identify the function to differentiate.\newlineWe are given the function f(x)=1−cos⁡(4x)1+cos⁡(4x)f(x) = \sqrt{\frac{1 - \cos(4x)}{1 + \cos(4x)}}. We need to find its derivative with respect to xx.
  2. Apply Chain Rule: Apply the chain rule for differentiation.\newlineThe chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is the square root and the inner function is the quotient (1−cos⁡(4x))/(1+cos⁡(4x))(1 - \cos(4x)) / (1 + \cos(4x)).
  3. Differentiate Outer Function: Differentiate the outer function.\newlineThe derivative of u\sqrt{u} with respect to uu is 12u\frac{1}{2\sqrt{u}}. We will apply this to the inner function later.
  4. Differentiate Inner Function: Differentiate the inner function.\newlineThe inner function is a quotient, so we need to use the quotient rule: (v′⋅u−u′⋅v)/v2(v' \cdot u - u' \cdot v) / v^{2}, where u=1−cos⁡(4x)u = 1 - \cos(4x) and v=1+cos⁡(4x)v = 1 + \cos(4x).
  5. Differentiate uu and vv: Differentiate uu and vv with respect to xx.
    u′=u' = derivative of (1−cos⁡(4x))=4sin⁡(4x)(1 - \cos(4x)) = 4\sin(4x)
    v′=v' = derivative of (1+cos⁡(4x))=−4sin⁡(4x)(1 + \cos(4x)) = -4\sin(4x)
  6. Apply Quotient Rule: Apply the quotient rule.\newlineUsing the derivatives from Step 55, we get:\newline(u′⋅v−u⋅v′)/v2=(4sin⁡(4x)⋅(1+cos⁡(4x))−(1−cos⁡(4x))⋅(−4sin⁡(4x)))/(1+cos⁡(4x))2(u' \cdot v - u \cdot v') / v^2 = (4\sin(4x) \cdot (1 + \cos(4x)) - (1 - \cos(4x)) \cdot (-4\sin(4x))) / (1 + \cos(4x))^2
  7. Simplify Expression: Simplify the expression.\newlineSimplify the numerator:\newline4sin⁡(4x)⋅(1+cos⁡(4x))+4sin⁡(4x)⋅(1−cos⁡(4x))=4sin⁡(4x)+4sin⁡(4x)cos⁡(4x)+4sin⁡(4x)−4sin⁡(4x)cos⁡(4x)=8sin⁡(4x)4\sin(4x) \cdot (1 + \cos(4x)) + 4\sin(4x) \cdot (1 - \cos(4x)) = 4\sin(4x) + 4\sin(4x)\cos(4x) + 4\sin(4x) - 4\sin(4x)\cos(4x) = 8\sin(4x)\newlineThe denominator remains (1+cos⁡(4x))2(1 + \cos(4x))^2.\newlineSo, the derivative of the inner function is 8sin⁡(4x)(1+cos⁡(4x))2\frac{8\sin(4x)}{(1 + \cos(4x))^2}.
  8. Combine Derivatives: Combine the derivatives of the outer and inner functions.\newlineNow we multiply the derivative of the outer function by the derivative of the inner function:\newline121−cos⁡(4x)1+cos⁡(4x)\frac{1}{2\sqrt{\frac{1 - \cos(4x)}{1 + \cos(4x)}}} * 8sin⁡(4x)(1+cos⁡(4x))2\frac{8\sin(4x)}{(1 + \cos(4x))^2}
  9. Simplify Final Expression: Simplify the final expression.\newlineThe final derivative is:\newlinef′(x)=4sin⁡(4x)(1+cos⁡(4x))1−cos⁡(4x)1+cos⁡(4x)f'(x) = \frac{4\sin(4x)}{(1 + \cos(4x)) \sqrt{\frac{1 - \cos(4x)}{1 + \cos(4x)}}}

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