Question

Найти производную данной функции и упростить ее: y=tg(x/2) -ctg (x/2)

Answer 1

$y=tg \frac{x}{2} -ctg \frac{x}{2} \\\ y'= \frac{1}{2\cos^2 \frac{x}{2} } -(- \frac{1}{2\sin \frac{x}{2} } )= \frac{1}{2\cos^2 \frac{x}{2} } +\frac{1}{2\sin ^2\frac{x}{2} } = \\\ = \frac{\sin ^2\frac{x}{2}}{2\sin ^2\frac{x}{2}\cos^2 \frac{x}{2} } +\frac{\cos^2 \frac{x}{2}}{2\sin ^2\frac{x}{2}\cos^2 \frac{x}{2} } =\frac{\sin^2\frac{x}{2}+\cos^2\frac{x}{2}}{2\sin ^2\frac{x}{2}\cos^2 \frac{x}{2} } =\frac{2}{4\sin ^2\frac{x}{2}\cos^2 \frac{x}{2} } =\frac{2}{\sin ^2x }$

Answer 2

Finding the Derivative of the Given Function

To find the derivative of the given function y = tan(x/2) - cot(x/2), we can use the rules of differentiation. The derivative of tan(u) is sec^2(u)du and the derivative of cot(u) is -csc^2(u)du. Therefore, the derivative of the given function can be found using these rules.

The derivative of the given function is: y' = sec^2(x/2) * (1/2) - (-csc^2(x/2)) * (1/2)

Let's simplify this derivative.

Simplifying the Derivative

Simplifying the derivative, we get: y' = (1/2)sec^2(x/2) + (1/2)csc^2(x/2)

This is the simplified form of the derivative of the given function.

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