Find the derivative of ^-1x + ^-1x, x, and ^-1sqrt(1+x2) + ^-1sqrt(1+x2) with respect to x.

Here are the solutions to the problems:

Question 2:
We need to find the derivative of $\sec^{-1}x + \csc^{-1}x$ with respect to $x$.

Step 1: Recall the inverse trigonometric identity $\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$.

Step 2: Substitute the identity into the expression.

$$\frac{d}{dx}(\sec^{-1}x + \csc^{-1}x) = \frac{d}{dx}\left(\frac{\pi}{2}\right)$$

Step 3: Differentiate the constant.

$$\frac{d}{dx}\left(\frac{\pi}{2}\right) = 0$$

The correct option is (b).
The final answer is $\boxed{\text{0}}$.

Question 3:
We need to find the derivative of $\sec x$ with respect to $x$.

Step 1: Recall the standard derivative of $\sec x$.

$$\frac{d}{dx}(\sec x) = \sec x \tan x$$

The correct option is (a).
The final answer is $\boxed{\sec x \tan x}$.

Question 4:
We need to find the derivative of $\tan^{-1}\sqrt{1+x^2} + \cot^{-1}\sqrt{1+x^2}$ with respect to $x$.

Step 1: Recall the inverse trigonometric identity $\tan^{-1}u + \cot^{-1}u = \frac{\pi}{2}$. Here, $u = \sqrt{1+x^2}$.

Step 2: Substitute the identity into the expression.

$$\frac{d}{dx}(\tan^{-1}\sqrt{1+x^2} + \cot^{-1}\sqrt{1+x^2}) = \frac{d}{dx}\left(\frac{\pi}{2}\right)$$

Step 3: Differentiate the constant.

$$\frac{d}{dx}\left(\frac{\pi}{2}\right) = 0$$

The correct option is (d).
The final answer is $\boxed{\text{0}}$.

Question 5:
We need to find $\frac{d}{d(3x)}(2^x)$. This is a derivative with respect to $3x$.

Step 1: Let $y = 2^x$ and $u = 3x$. We want to find $\frac{dy}{du}$.

Step 2: Find $\frac{dy}{dx}$ and $\frac{du}{dx}$.

$$\frac{dy}{dx} = \frac{d}{dx}(2^x) = 2^x \ln 2$$ $$\frac{du}{dx} = \frac{d}{dx}(3x) = 3$$

Step 3: Use the chain rule $\frac{dy}{du} = \frac{dy/dx}{du/dx}$.

$$\frac{dy}{du} = \frac{2^x \ln 2}{3}$$

The result is $\frac{1}{3} 2^x \ln 2$, which does not match options (a), (b), or (c).
The correct option is (d).
The final answer is $\boxed{\frac{1}{3} 2^x \ln 2}$.

Question 6:
Given $y = \tan^{-1}\left[\frac{\cos x}{1+\sin x}\right]$, find $\frac{dy}{dx}$.

Step 1: Simplify the expression inside the $\tan^{-1}$ function.

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