Find out the derivtive of (cos x)/x using the first principle of derivative

To find the derivative of the function \( f(x) = \frac{\cos x}{x} \) using the first principle of derivatives, also known as the definition of the derivative, we start with the basic formula:

Understanding the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as:

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

This formula gives us the slope of the tangent line to the curve at the point \( x \). For our function, we will apply this definition step by step.

Step 1: Set Up the Expression

We need to compute \( f(x + h) \) and \( f(x) \):

• \( f(x) = \frac{\cos x}{x} \)
• \( f(x + h) = \frac{\cos(x + h)}{x + h} \)

Step 2: Substitute into the Derivative Formula

Now, we substitute these into the derivative formula:

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

Step 3: Combine the Fractions

To simplify the expression, we need a common denominator:

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

Step 4: Simplifying the Numerator

Now, we focus on simplifying the numerator:

\( x \cos(x + h) - (x + h) \cos x = x \cos(x + h) - x \cos x - h \cos x \)

This can be rewritten as:

\( x (\cos(x + h) - \cos x) - h \cos x \)

Step 5: Applying the Limit

Now, we can break this limit into two parts:

\( f'(x) = \lim_{h \to 0} \left( \frac{x (\cos(x + h) - \cos x)}{h (x + h)} - \frac{\cos x}{x + h} \right) \end{align*} \)

Step 6: Using the Cosine Difference Formula

Recall that the limit of \( \frac{\cos(x + h) - \cos x}{h} \) as \( h \to 0 \) is equal to \( -\sin x \). Thus, we can substitute this into our expression:

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

Step 7: Finalizing the Derivative

As \( h \) approaches 0, the second term simplifies to \( -\frac{\cos x}{x} \). Therefore, we have:

\( f'(x) = -\frac{x \sin x}{x^2} - \frac{\cos x}{x} \)

Final Expression

Combining these terms gives us:

\( f'(x) = -\frac{\sin x}{x} - \frac{\cos x}{x^2} \)

Thus, the derivative of \( f(x) = \frac{\cos x}{x} \) is:

\( f'(x) = -\frac{\sin x}{x} - \frac{\cos x}{x^2} \)

This method illustrates how we can derive the derivative using the first principles, emphasizing the importance of limits and simplification in calculus.