y=ln tan((pi)/(4)+(x)/(2))

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Identify Outer Function: Identify the outer function and its derivative. The outer function is the natural logarithm ln(u), where u is an inner function. The derivative of ln(u) with respect to u is 1/u.Identify Inner Function: Identify the inner function and its derivative. The inner function is tan((π/4) + (x/2)). To find its derivative, we will use the chain rule. The derivative of tan(v) with respect to v is sec^2(v), where v is (π/4) + (x/2).Apply Chain Rule: Apply the chain rule. The chain rule states that the derivative of a composite function ln(tan(v)) with respect to x is the derivative of ln(u) with respect to u times the derivative of u with respect to x. So we have: dy/dx = d/dx[ln(tan((π/4) + (x/2)))] = (1/tan((π/4) + (x/2))) * d/dx[tan((π/4) + (x/2))]Calculate Inner Function Derivative: Calculate the derivative of the inner function. We need to find the derivative of tan((π/4) + (x/2)) with respect to x. Using the derivative of tan(v) which is sec^2(v), we get: d/dx[tan((π/4) + (x/2))] = sec^2((π/4) + (x/2)) * d/dx[(π/4) + (x/2)] Since the derivative of (π/4) with respect to x is 0 and the derivative of (x/2) with respect to x is 1/2, we have: d/dx[tan((π/4) + (x/2))] = sec^2((π/4) + (x/2)) * (1/2)Combine Results: Combine the results from Step 3 and Step 4. Now we multiply the derivative of the outer function by the derivative of the inner function to get the final derivative: dy/dx = (1/tan((π/4) + (x/2))) * sec^2((π/4) + (x/2)) * (1/2)Simplify Expression: Simplify the expression. We know that sec^2(v) is 1/cos^2(v) and tan(v) is sin(v)/cos(v). Therefore, we can simplify the expression by multiplying the sec^2 term by the reciprocal of the tan term: dy/dx = (1/(sin((π/4) + (x/2))/cos((π/4) + (x/2)))) * (1/cos^2((π/4) + (x/2))) * (1/2)Further Simplify Expression: Further simplify the expression. We can simplify the expression by multiplying the cos term in the numerator with the cos^2 term in the denominator: dy/dx = (cos((π/4) + (x/2))/sin((π/4) + (x/2))) * (1/cos^2((π/4) + (x/2))) * (1/2) dy/dx = (1/sin((π/4) + (x/2))) * (1/cos((π/4) + (x/2))) * (1/2) dy/dx = (1/sin((π/4) + (x/2))) * (sec((π/4) + (x/2))) * (1/2)Recognize Cosecant Function: Recognize that 1/sin(v) is csc(v). We can rewrite the expression using the cosecant function: dy/dx = csc((π/4) + (x/2)) * sec((π/4) + (x/2)) * (1/2)Finalize Derivative: Finalize the derivative. The final derivative of y with respect to x is: dy/dx = (1/2) * csc((π/4) + (x/2)) * sec((π/4) + (x/2))

$y=\ln\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)$

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$y=\ln\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)$