Derivative of 1/√(1-x) Using the First Principle

In calculus, the derivative of a function is a measure of how the function changes as its input changes. The first principle of derivatives is one of the most fundamental methods to find the derivative of a given function. In this article, we will explore the derivative of the function 1/√(1-x) using the first principle.

The First Principle of Derivatives

The first principle of derivatives, also known as the definition of a derivative, is given by the limit:

(frac{dy}{dx} lim_{h to 0} frac{f(x h) - f(x)}{h})

Applying the First Principle to 1/√(1-x)

Let's find the derivative of the function 1/√(1-x) using the first principle. The function in question is:

(f(x) frac{1}{sqrt{1-x}})

The first step is to calculate:

(f(x h) - f(x))

Substituting the function, we get:

(f(x h) frac{1}{sqrt{1-(x h)}} quad text{and} quad f(x) frac{1}{sqrt{1-x}})

Thus,

(f(x h) - f(x) frac{1}{sqrt{1-(x h)}} - frac{1}{sqrt{1-x}})

Now, dividing by h:

(frac{f(x h) - f(x)}{h} frac{1}{h} left(frac{1}{sqrt{1-(x h)}} - frac{1}{sqrt{1-x}}right))

Applying the Limit

We need to compute the limit as h approaches 0:

(lim_{h to 0} frac{1}{h} left(frac{1}{sqrt{1-(x h)}} - frac{1}{sqrt{1-x}}right))

To simplify the expression, we rationalize the numerator:

(frac{1}{h} left(frac{1}{sqrt{1-(x h)}} - frac{1}{sqrt{1-x}}right) cdot frac{sqrt{1-(x h)} sqrt{1-x}}{sqrt{1-(x h)} sqrt{1-x}})

This gives us:

(lim_{h to 0} frac{sqrt{1-x} - sqrt{1-(x h)}}{h sqrt{1-(x h)} sqrt{1-x}})

To simplify further, we use the identity ((a - b)(a b) a^2 - b^2), where (a sqrt{1-x}) and (b sqrt{1-(x h)}):

(lim_{h to 0} frac{(sqrt{1-x} - sqrt{1-(x h)})(sqrt{1-x} sqrt{1-(x h)})}{h sqrt{1-(x h)} sqrt{1-x}(sqrt{1-x} sqrt{1-(x h)})})

Simplifying the numerator:

(lim_{h to 0} frac{(1-x) - (1-(x h))}{h sqrt{1-(x h)} sqrt{1-x}(sqrt{1-x} sqrt{1-(x h)})})

Which simplifies to:

(lim_{h to 0} frac{h}{h sqrt{1-(x h)} sqrt{1-x}(sqrt{1-x} sqrt{1-(x h)})})

Cancelling h in the numerator and denominator:

(lim_{h to 0} frac{1}{sqrt{1-(x h)} sqrt{1-x}(sqrt{1-x} sqrt{1-(x h)})})

Now, taking the limit as h approaches 0:

(frac{1}{sqrt{1-x} sqrt{1-x}(sqrt{1-x} sqrt{1-x})} frac{1}{2sqrt{1-x}sqrt{1-x}sqrt{1-x}} frac{1}{2(1-x)^{3/2}})

Therefore, the derivative of the function 1/√(1-x) is:

(frac{d}{dx} left(frac{1}{sqrt{1-x}}right) frac{1}{2(1-x)^{3/2}})

Summary

In this article, we used the first principle to compute the derivative of the function 1/√(1-x). The final result is (frac{1}{2(1-x)^{3/2}}). Understanding and applying the first principle is fundamental in calculus, providing a deep insight into the behavior of functions at specific points.