Introduction to Finding Derivatives Using the First Principle

Understanding the fundamental concept of derivatives is crucial in calculus. The derivatives of certain functions, such as the natural logarithmic function and the inverse tangent function, can be found using the first principle of derivatives. This method provides a foundational approach to calculating derivatives by starting with the definition of the derivative as the rate of change of a function with respect to its variable.

The Derivative of Natural Logarithmic Function ln x

The function ln x (natural logarithm of x) is a logarithmic function with base e. To find its derivative, we can use the first principle of derivatives:

[ frac{d}{dx} ln x  lim_{Delta x to 0} frac{ln(x   Delta x) - ln x}{Delta x} ]

Using properties of logarithms, the expression can be simplified as follows:

[ frac{d}{dx} ln x  lim_{Delta x to 0} ln left( left(1   frac{Delta x}{x} right)^{1/Delta x} right)^{1/x} ]

Here, the term inside the logarithm is of the form (1 n)^{1/n}, where n frac{Delta x}{x}. As Delta x to 0, this term converges to e. Thus, the derivative simplifies to:

[ frac{d}{dx} ln x  frac{1}{x} ln e  frac{1}{x} ]

Hence, the derivative of ln x is:

[ frac{d}{dx} ln x  frac{1}{x} quad text{for} quad x 
eq 0 ]

The Derivative of Inverse Tangent Function tan^{-1} x

The inverse tangent function, denoted as tan^{-1} x, represents the angle whose tangent is x. To find the derivative of this function, we use the first principle and the properties of the tangent function:

[ text{Let} quad y  tan^{-1} x ][ text{Then} quad x  tan y ][ frac{d}{dx} x  frac{d}{dx} tan y ][ 1  sec^2 y frac{dy}{dx} ]

Using the identity sec^2 y 1 tan^2 y, we get:

[ 1  (1   tan^2 y) frac{dy}{dx} ][ 1  (1   x^2) frac{dy}{dx} ][ frac{dy}{dx}  frac{1}{1   x^2} ]

Thus, the derivative of tan^{-1} x is:

[ frac{d}{dx} tan^{-1} x  frac{1}{1   x^2} ]

Conclusion and Applications

Understanding the derivative of these functions is crucial in various mathematical and real-world applications, such as physics, engineering, and finance. These derivatives are used to analyze the behavior of functions and solve optimization problems, among others.

If you need further examples or have additional questions related to these derivatives, feel free to ask!