The slope of the tangent to the curve
`y=ln(cosx)" at "x=(3pi)/(4)" is "`

Options

A. 1
B. -1
C. `l nsqrt(2)`
D. `(1)/(sqrt(2))`

Answer

To find the slope of the tangent to the curve \( y = \ln(\cos x) \) at \( x = \frac{3\pi}{4} \), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = \ln(\cos x) \]
To find the slope of the tangent, we need to differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}[\ln(\cos x)] \]

### Step 2: Apply the chain rule
Using the chain rule for differentiation, we have:
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot \frac{d}{dx}[\cos x] \]
We know that the derivative of \( \cos x \) is \( -\sin x \). Therefore, we can substitute this into our equation:
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) = -\frac{\sin x}{\cos x} \]

### Step 3: Simplify the expression
The expression simplifies to:
\[ \frac{dy}{dx} = -\tan x \]

### Step 4: Evaluate the derivative at \( x = \frac{3\pi}{4} \)
Now we need to find the slope at \( x = \frac{3\pi}{4} \):
\[ \frac{dy}{dx} \bigg|_{x=\frac{3\pi}{4}} = -\tan\left(\frac{3\pi}{4}\right) \]

### Step 5: Calculate \( \tan\left(\frac{3\pi}{4}\right) \)
Using the property of tangent:
\[ \tan\left(\frac{3\pi}{4}\right) = \tan\left(\pi - \frac{\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) \]
We know that \( \tan\left(\frac{\pi}{4}\right) = 1 \), so:
\[ \tan\left(\frac{3\pi}{4}\right) = -1 \]

### Step 6: Substitute back to find the slope
Now substituting back:
\[ \frac{dy}{dx} \bigg|_{x=\frac{3\pi}{4}} = -(-1) = 1 \]

### Conclusion
The slope of the tangent to the curve \( y = \ln(\cos x) \) at \( x = \frac{3\pi}{4} \) is:
\[ \boxed{1} \]

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