Differentiate log sin x by first principle.

To differentiate the function \( \log(\sin x) \) using the first principle of calculus, we start by recalling what the first principle entails. The first principle, also known as the definition of the derivative, states that the derivative of a function \( f(x) \) at a point \( x \) is given by the limit:

Understanding the First Principle of Derivatives

The derivative \( f'(x) \) can be expressed as:

\( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

In our case, the function \( f(x) = \log(\sin x) \). Therefore, we need to evaluate:

Setting Up the Limit

We substitute \( f(x) \) into the limit definition:

\( f'(x) = \lim_{h \to 0} \frac{\log(\sin(x+h)) - \log(\sin x)}{h} \)

Using the properties of logarithms, specifically that \( \log a - \log b = \log\left(\frac{a}{b}\right) \), we can rewrite this as:

\( f'(x) = \lim_{h \to 0} \frac{\log\left(\frac{\sin(x+h)}{\sin x}\right)}{h} \)

Applying the Limit

Now, we can apply the limit. To simplify \( \frac{\sin(x+h)}{\sin x} \), we can use the sine addition formula:

• Recall that \( \sin(x+h) = \sin x \cos h + \cos x \sin h \).

Thus, we have:

\( \frac{\sin(x+h)}{\sin x} = \frac{\sin x \cos h + \cos x \sin h}{\sin x} = \cos h + \frac{\cos x \sin h}{\sin x} \)

As \( h \) approaches 0, \( \cos h \) approaches 1, and \( \frac{\sin h}{h} \) approaches 1. Therefore, we can evaluate the limit:

Final Limit Evaluation

Substituting back into our limit, we get:

\( f'(x) = \lim_{h \to 0} \frac{\log\left(1 + \frac{\cos x \sin h}{\sin x}\right)}{h} \)

Using the fact that \( \log(1 + u) \approx u \) for small \( u \), we can approximate:

\( \log\left(1 + \frac{\cos x \sin h}{\sin x}\right) \approx \frac{\cos x \sin h}{\sin x} \)

Thus, our limit becomes:

\( f'(x) = \lim_{h \to 0} \frac{\frac{\cos x \sin h}{\sin x}}{h} = \frac{\cos x}{\sin x} \cdot \lim_{h \to 0} \frac{\sin h}{h} = \frac{\cos x}{\sin x} \cdot 1 = \cot x \)

Final Result

Therefore, the derivative of \( \log(\sin x) \) is:

\( \frac{d}{dx} \log(\sin x) = \cot x \)

This process illustrates how we can use the first principle of derivatives to find the derivative of a logarithmic function involving sine, demonstrating both the power of limits and the properties of logarithms in calculus.