** Final Answer

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** Step-by-step Solution **

** Specify the solving method

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To derive the function $\tan\left(x+1\right)$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

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Apply natural logarithm to both sides of the equality

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Apply logarithm properties to both sides of the equality

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Derive both sides of the equality with respect to $x$

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The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

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The derivative of the linear function is equal to $1$

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The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

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The derivative of a sum of two or more functions is the sum of the derivatives of each function

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The derivative of the constant function ($1$) is equal to zero

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The derivative of the linear function is equal to $1$

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Multiply the fraction and term

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Simplify $\frac{\sec\left(x+1\right)^2}{\tan\left(x+1\right)}$ by applying trigonometric identities

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Multiply both sides of the equation by $y$

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Substitute $y$ for the original function: $\tan\left(x+1\right)$

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The derivative of the function results in

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Simplify the derivative

** Final Answer

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