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$y=\arctan(\tan x)$

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Find derivatives of inverse trigonometric functions

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Q.

y=\arctan(\tan x)

Recognize function notation: step $_1$ : Recognize that $\text{arctg}$ is another notation for $\text{arctan}$ , which is the inverse tangent function.
Apply chain rule for differentiation: step_2: Use the chain rule for differentiation, which 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.
Find derivative of outer function: step_3: Find the derivative of the outer function, which is $\arctan(u)$ , where $u = \tan(x)$ . The derivative of $\arctan(u)$ with respect to $u$ is $\frac{1}{1+u^2}$ .
Find derivative of inner function: step extunderscore{} $4$ : Find the derivative of the inner function, which is $\tan(x)$ . The derivative of $\tan(x)$ with respect to $x$ is $\sec^2(x)$ .
Apply chain rule: step_5: Apply the chain rule. The derivative of $y$ with respect to $x$ is the derivative of $\arctan(\tan(x))$ with respect to $\tan(x)$ times the derivative of $\tan(x)$ with respect to $x$ . $\frac{dy}{dx} = \left(\frac{1}{1+(\tan(x))^2}\right) \cdot \sec^2(x)$
Simplify expression: step_6: Simplify the expression. Since $\tan x = \frac{\sin x}{\cos x}$ , and $\sec x = \frac{1}{\cos x}$ , $\sec^2(x) = \frac{1}{\cos^2(x)}$ . $\newline$ $\frac{dy}{dx} = \left(\frac{1}{1+\left(\frac{\sin x}{\cos x}\right)^2}\right) * \left(\frac{1}{\cos^2(x)}\right)$
Simplify further: step_7: Simplify further by multiplying the numerator and denominator of the first fraction by $\cos^2(x)$ to get rid of the complex fraction. $\newline$ $\frac{dy}{dx} = \left(\frac{\cos^2(x)}{\cos^2(x)+\sin^2(x)}\right) * \left(\frac{1}{\cos^2(x)}\right)$
Recognize trigonometric identity: step_8: Recognize that $\cos^2(x) + \sin^2(x) = 1$ , which is a basic trigonometric identity. $\newline$ $\frac{dy}{dx} = \left(\frac{\cos^2(x)}{1}\right) * \left(\frac{1}{\cos^2(x)}\right)$
Simplify final expression: step_9: Simplify the expression by canceling out $\cos^2(x)$ in the numerator and denominator. $\frac{dy}{dx} = 1$

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