Differentiate `f(x)=tan 2x` by first principle of differentiation.

Answer

To differentiate the function \( f(x) = \tan(2x) \) using the first principle of differentiation, we will follow these steps:

### Step 1: Write the definition of the derivative
The first principle of differentiation states that:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 2: Substitute the function into the formula
Substituting \( f(x) = \tan(2x) \):
\[ f'(x) = \lim_{h \to 0} \frac{\tan(2(x+h)) - \tan(2x)}{h} \]
This simplifies to:
\[ f'(x) = \lim_{h \to 0} \frac{\tan(2x + 2h) - \tan(2x)}{h} \]

### Step 3: Use the tangent subtraction formula
Using the tangent subtraction formula, we can express the difference of tangents:
\[ \tan(A) - \tan(B) = \frac{\sin(A - B)}{\cos(A)\cos(B)} \]
Thus, we can rewrite:
\[ f'(x) = \lim_{h \to 0} \frac{\sin(2h)}{\cos(2x + 2h)\cos(2x)} \cdot \frac{1}{h} \]

### Step 4: Rewrite the limit
Now we can rewrite the limit:
\[ f'(x) = \lim_{h \to 0} \frac{\sin(2h)}{2h} \cdot \frac{2}{\cos(2x + 2h)\cos(2x)} \]
As \( h \to 0 \), \( \frac{\sin(2h)}{2h} \to 1 \).

### Step 5: Evaluate the limit
Now we can evaluate the limit:
\[ f'(x) = 1 \cdot \frac{2}{\cos(2x)\cos(2x)} = \frac{2}{\cos^2(2x)} \]

### Step 6: Use the secant identity
Since \( \frac{1}{\cos^2(2x)} = \sec^2(2x) \), we can write:
\[ f'(x) = 2 \sec^2(2x) \]

### Conclusion
Thus, the derivative of \( f(x) = \tan(2x) \) is:
\[ f'(x) = 2 \sec^2(2x) \]

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