Derivative of ( sqrt{x} - frac{1}{x} ) Using Chain, Product, Quotient, and Sum Rules

In calculus, understanding how to compute the derivative of complex expressions is a fundamental concept. This article will walk you through the step-by-step process of finding the derivative of ( sqrt{x} - frac{1}{x} ) using the chain, product, quotient, and sum rules. We will also provide an explanation of each of these rules for better comprehension.

The Chain Rule

The chain rule is a technique used to find the derivative of a composite function. In simpler terms, it is used when a function is composed of another function, such as ( f(g(x)) ). The chain rule states that (frac{d}{dx}f(g(x)) f'(g(x))g'(x)).

Problems Facing the Given Derivative

The initial problem posed is to find the derivative of ( sqrt{x} - frac{1}{x} ). To simplify this problem, let's rewrite it as: ( y x - frac{1}{x^{1/2}} ).

Applying the Sum, Product, Quotient, and Chain Rules

Let's break down the expression ( y x - frac{1}{x^{1/2}} ).

Sum Rule

The sum rule states that the derivative of the sum of two functions is the sum of the derivatives of the individual functions. Therefore, we can rewrite the derivative of ( y x - frac{1}{x^{1/2}} ) as:

( frac{dy}{dx} frac{d}{dx}x - frac{d}{dx}frac{1}{x^{1/2}} )

Quotient Rule

The quotient rule is used to find the derivative of a function that is the ratio of two functions, ( frac{f(x)}{g(x)} ). The formula is: ( frac{d}{dx}frac{f(x)}{g(x)} frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} )

Let’s apply this rule to the second term, ( frac{1}{x^{1/2}} ):

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

Chain Rule

For the term ( sqrt{x} ), we can use the chain rule. Recognize that ( sqrt{x} x^{1/2} ), and apply the chain rule:

( frac{d}{dx}x^{1/2} frac{1}{2}x^{-1/2} )

Combining these results:

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

Final Simplification

The final expression for the derivative is:

( frac{dy}{dx} 1 frac{1}{2x^{3/2}} )

Combining the Rules

Let's verify the process step-by-step:

Starting with the original function: ( y x - frac{1}{x^{1/2}} ) Using the sum rule: ( frac{dy}{dx} frac{d}{dx}x - frac{d}{dx}frac{1}{x^{1/2}} ) Applying the quotient rule: ( frac{d}{dx}frac{1}{x^{1/2}} -frac{1}{2}x^{-3/2} ) Using the power rule: ( frac{d}{dx}x^{1/2} frac{1}{2}x^{-1/2} ) Combining the results: ( frac{dy}{dx} 1 frac{1}{2x^{3/2}} )

Conclusion

The derivative of ( sqrt{x} - frac{1}{x} ) is ( 1 frac{1}{2x^{3/2}} ). This process highlights the power of using the chain, product, quotient, and sum rules in calculus to solve complex problems.

Summary of the Rules

Sum Rule: ( frac{d}{dx}(f(x) g(x)) frac{df}{dx} frac{dg}{dx} )

Quotient Rule: ( frac{d}{dx}left(frac{f(x)}{g(x)}right) frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} )

Chain Rule: ( frac{d}{dx}f(g(x)) f'(g(x))g'(x) )

Power Rule: ( frac{d}{dx}x^n nx^{n-1} )

Understanding and applying these rules is crucial for solving more complex calculus problems.