Find `(dy)/(dx)` if `y=tanx`.

Explanation

To find \(\frac{dy}{dx}\) for the function \(y = \tan x\), we can use the quotient rule of differentiation. Here’s the step-by-step solution:

### Step 1: Rewrite \(\tan x\)
We know that:
\[ y = \tan x = \frac{\sin x}{\cos x} \]

### Step 2: Apply the Quotient Rule
The quotient rule states that if \(y = \frac{u}{v}\), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
In our case, \(u = \sin x\) and \(v = \cos x\).

### Step 3: Differentiate \(u\) and \(v\)
Now we need to find \(\frac{du}{dx}\) and \(\frac{dv}{dx}\):
- \(\frac{du}{dx} = \frac{d}{dx}(\sin x) = \cos x\)
- \(\frac{dv}{dx} = \frac{d}{dx}(\cos x) = -\sin x\)

### Step 4: Substitute into the Quotient Rule
Now substituting \(u\), \(v\), \(\frac{du}{dx}\), and \(\frac{dv}{dx}\) into the quotient rule formula:
\[ \frac{dy}{dx} = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \frac{dy}{dx} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \]

### Step 6: Use the Pythagorean Identity
Using the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\), we can simplify further:
\[ \frac{dy}{dx} = \frac{1}{\cos^2 x} \]

### Step 7: Write the Final Result
Finally, we can express \(\frac{1}{\cos^2 x}\) as \(\sec^2 x\):
\[ \frac{dy}{dx} = \sec^2 x \]

### Summary
Thus, the derivative of \(y = \tan x\) is:
\[ \frac{dy}{dx} = \sec^2 x \]