Final Answer
$\sec\left(x+1\right)^2$
Step-by-step Solution
Problem to solve:
$\frac{d}{dx}\left(\tan\left(x+1\right)\right)$
Solving method
1
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 ${f'(x) = sec^2(x)\cdot D_x(x)}$
$\sec\left(x+1\right)^2\frac{d}{dx}\left(x+1\right)$
2
The derivative of a sum of two functions is the sum of the derivatives of each function
$\sec\left(x+1\right)^2\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(1\right)\right)$
$\sec\left(x+1\right)^2\left(\frac{d}{dx}\left(x\right)+0\right)$
$x+0=x$, where $x$ is any expression
$\sec\left(x+1\right)^2\frac{d}{dx}\left(x\right)$
3
The derivative of the constant function ($1$) is equal to zero
$\sec\left(x+1\right)^2\frac{d}{dx}\left(x\right)$
$1\sec\left(x+1\right)^2$
Any expression multiplied by $1$ is equal to itself
$\sec\left(x+1\right)^2$
4
The derivative of the linear function is equal to $1$
$\sec\left(x+1\right)^2$
Final Answer
$\sec\left(x+1\right)^2$