Question

Differentiate the function \(y=\frac{1}{\sqrt{x}}\).

Short answer

Question: Find the derivative of the function \(y=\frac{1}{\sqrt{x}}\). Answer: The derivative of the function \(y=\frac{1}{\sqrt{x}}\) is \(y'=-\frac{1}{2x^{\frac{3}{2}}}\).

Step-by-step solution

Rewrite function with negative exponent

Rewrite \(\frac{1}{\sqrt{x}}\) as \(x^{-\frac{1}{2}}\). The function becomes \(y=x^{-\frac{1}{2}}\).

Apply the power rule

To differentiate the function, apply the power rule, which states that \((x^n)' = nx^{n-1}\). In this case, \(n = -\frac{1}{2}\).

Differentiate the function

Differentiate the function: \((x^{-\frac{1}{2}})' = -\frac{1}{2}x^{-\frac{1}{2}-1}\).

Simplify the derivative

Simplify the derivative: \(-\frac{1}{2}x^{-\frac{3}{2}}\). Now, rewrite the derivative with a positive exponent in the denominator: \(y'=-\frac{1}{2x^{\frac{3}{2}}}\). The derivative of the function \(y=\frac{1}{\sqrt{x}}\) is \(y'=-\frac{1}{2x^{\frac{3}{2}}}\).

Key concepts

Power Rule

The power rule is a fundamental tool in calculus for differentiation. It provides a quick and easy method to find the derivative of powers of a variable. The power rule states that if you have a function of the form \(x^n\), its derivative is \(nx^{n-1}\). This rule applies to any real number exponent \(n\).
For example:

• If \(f(x) = x^3\), then \(f'(x) = 3x^{2}\).
• If \(f(x) = x^{-1}\), then \(f'(x) = -1x^{-2}\).

In our exercise, the function \(y = x^{-rac{1}{2}}\) was differentiated using the power rule. This rule simplifies differentiation, especially when dealing with polynomials and functions that can be expressed as a power of a variable.

Negative Exponents

Negative exponents can be a bit tricky, but they follow a simple rule: \(x^{-n} = \frac{1}{x^n}\). This means you can rewrite negative exponents as fractions, which is especially useful when simplifying derivatives.
In the original function \(y = \frac{1}{\sqrt{x}}\), rewriting it as \(y = x^{-rac{1}{2}}\) allows us to use the power rule directly.
Remember:

• A negative exponent moves the base to the denominator.
• Converting to negative exponents can make differentiation easier.

Once you differentiate, you might want to convert back from a negative exponent to a fraction to express the result in its simplest form.

Functions

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In calculus, functions are fundamental objects of study. They can be as simple as a linear function or as complex as a multivariable exponential function.
The function in this exercise is \(y = \frac{1}{\sqrt{x}}\). This function defines a particular type of curve. By differentiating it, you can determine how steep or flat the curve is at any point.
The process of differentiation transforms this function into another function, called a derivative, which provides the slope at each point of the original function's graph. Understanding the nature of different types of functions and their differentiability is crucial in calculus.

Derivative Simplification

Once you've found the derivative, the next step is often simplification. Simplifying a derivative involves rewriting it in a form that is easier to interpret or work with. The final solution for the derivative in this exercise was simplified as \(-\frac{1}{2x^{\frac{3}{2}}}\).
Simplification tips:

• Change negative exponents to fractions to avoid confusion.
• Combine like terms when possible, to make the expression as simple as possible.

In this problem, simplifying the derivative helped present the solution in a clear format that can be easily used for further mathematical operations or interpretations.