`tan^-1""((3-5x)/(1+15x))`

Answer

To solve the problem \( y = \tan^{-1}\left(\frac{3 - 5x}{1 + 15x}\right) \), we can use the formula for the tangent of the difference of two angles. The formula states that:

\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x - y}{1 + xy}\right) \]

In this case, we can identify \( x = 3 \) and \( y = 5x \). Thus, we can rewrite our expression as:

\[ y = \tan^{-1}(3) - \tan^{-1}(5x) \]

Now, we will differentiate \( y \) with respect to \( x \).

### Step 1: Differentiate \( y \)

Using the derivative of \( \tan^{-1}(u) \), which is given by:

\[ \frac{d}{dx} \tan^{-1}(u) = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \]

we can differentiate \( y \):

\[ \frac{dy}{dx} = \frac{d}{dx} \left( \tan^{-1}(3) - \tan^{-1}(5x) \right) \]

Since \( \tan^{-1}(3) \) is a constant, its derivative is zero. Therefore, we only need to differentiate \( -\tan^{-1}(5x) \):

\[ \frac{dy}{dx} = 0 - \frac{d}{dx} \tan^{-1}(5x) \]

### Step 2: Apply the derivative formula

Now we apply the derivative formula:

\[ \frac{dy}{dx} = -\frac{1}{1 + (5x)^2} \cdot \frac{d}{dx}(5x) \]

### Step 3: Differentiate \( 5x \)

The derivative of \( 5x \) with respect to \( x \) is \( 5 \):

\[ \frac{d}{dx}(5x) = 5 \]

### Step 4: Substitute back into the derivative

Substituting this back into our expression for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = -\frac{1}{1 + 25x^2} \cdot 5 \]

### Step 5: Simplify the expression

Thus, we have:

\[ \frac{dy}{dx} = -\frac{5}{1 + 25x^2} \]

### Final Answer

The final result for the derivative is:

\[ \frac{dy}{dx} = -\frac{5}{1 + 25x^2} \]

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