Find derivative of tan(x+4)

The correct option is sec²(x+4).

To find the derivative of the function $f\left(x\right)=tan(x+4)$, we apply the chain rule of differentiation.

The chain rule states that if we have a composite function $y=g\left(u\right)$ where $u=h\left(x\right)$, then the derivative of $y$ with respect to $x$ is given by:
$$\frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx}$$

Let us define the inner function as:
$u=x+4$

Then the outer function becomes:
$y=tan\left(u\right)$

Now, we differentiate the outer function with respect to $u$:
$$\frac{dy}{du}={sec}^2\left(u\right)$$

Next, we differentiate the inner function with respect to $x$:
$$\frac{du}{dx}=\frac{d}{dx}(x+4)=1+0=1$$

Finally, substituting these values back into the chain rule formula gives:
$$\frac{dy}{dx}={sec}^2\left(u\right)·1$$

Replacing $u$ with $x+4$ yields:
$$\frac{dy}{dx}={sec}^2(x+4)$$