Find the derivative of tan 5x using first principle of differentiation.

Answer

To find the derivative of \( \tan(5x) \) using the first principle of differentiation, we will follow these steps:

### Step 1: Write the formula for the derivative using the first principle.
The first principle of differentiation states that:
\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Here, we will let \( f(x) = \tan(5x) \).

### Step 2: Substitute \( f(x) \) into the formula.
We need to evaluate \( f(x+h) \):
\[ f(x+h) = \tan(5(x+h)) = \tan(5x + 5h) \]
Now, substituting into the derivative formula gives:
\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{\tan(5x + 5h) - \tan(5x)}{h} \]

### Step 3: Use the tangent addition formula.
We will use the formula for \( \tan(A + B) \):
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Let \( A = 5x \) and \( B = 5h \):
\[ \tan(5x + 5h) = \frac{\tan(5x) + \tan(5h)}{1 - \tan(5x) \tan(5h)} \]
Substituting this back into our limit gives:
\[ \frac{dy}{dx} = \lim_{h \to 0} \frac{\frac{\tan(5x) + \tan(5h)}{1 - \tan(5x) \tan(5h)} - \tan(5x)}{h} \]

### Step 4: Simplify the expression.
Combine the terms in the numerator:
\[ = \lim_{h \to 0} \frac{\tan(5x) + \tan(5h) - \tan(5x)(1 - \tan(5x) \tan(5h))}{h(1 - \tan(5x) \tan(5h))} \]
This simplifies to:
\[ = \lim_{h \to 0} \frac{\tan(5h) + \tan^2(5x) \tan(5h)}{h(1 - \tan(5x) \tan(5h))} \]

### Step 5: Factor out \( \tan(5h) \).
We can factor out \( \tan(5h) \):
\[ = \lim_{h \to 0} \frac{\tan(5h)(1 + \tan^2(5x))}{h(1 - \tan(5x) \tan(5h))} \]

### Step 6: Apply the limit.
Using the limit property \( \lim_{h \to 0} \frac{\tan(5h)}{h} = 5 \):
\[ = \lim_{h \to 0} \frac{5(1 + \tan^2(5x))}{1 - \tan(5x) \cdot 0} = 5(1 + \tan^2(5x)) \]

### Step 7: Use the identity for secant.
Recall that \( 1 + \tan^2(\theta) = \sec^2(\theta) \):
\[ = 5 \sec^2(5x) \]

### Final Answer:
Thus, the derivative of \( \tan(5x) \) is:
\[ \frac{d}{dx}(\tan(5x)) = 5 \sec^2(5x) \]