Differentiate the following functions w.r.t. x:
`tan (4x -7)`

Answer

To differentiate the function \( y = \tan(4x - 7) \) with respect to \( x \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
Let:
- \( u = 4x - 7 \) (inner function)
- \( y = \tan(u) \) (outer function)

### Step 2: Differentiate the outer function
The derivative of \( y \) with respect to \( u \) is:
\[ \frac{dy}{du} = \sec^2(u) \]

### Step 3: Differentiate the inner function
Now, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(4x - 7) = 4 \]

### Step 4: Apply the chain rule
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \sec^2(u) \cdot 4 \]

### Step 5: Substitute back the value of \( u \)
Now, replace \( u \) with \( 4x - 7 \):
\[ \frac{dy}{dx} = 4 \sec^2(4x - 7) \]

### Final Answer
Thus, the derivative of \( y = \tan(4x - 7) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 4 \sec^2(4x - 7) \]

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