Problem 22 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(x)=\sqrt{\frac{1}{x^{3}}}$$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = -\frac{3}{2}x^{-\frac{5}{2}}\).

Step 1: Rewrite the function

First, express the function in a form that is easier to differentiate. The original function is given as \(f(x) = \sqrt{\frac{1}{x^3}}\). This can be rewritten using exponents: \(f(x) = (x^{-3})^{1/2}\). Simplifying further, we have \(f(x) = x^{-3/2}\).

Step 2: Apply the power rule for derivatives

The power rule for differentiation states that if \(f(x) = x^{n}\), then \(f'(x) = nx^{n-1}\). In this case, \(n = -\frac{3}{2}\), so we differentiate: \[f'(x) = -\frac{3}{2}x^{-\frac{3}{2} - 1}\] \[f'(x) = -\frac{3}{2}x^{-\frac{5}{2}}\].

Step 3: Simplify the derivative expression

Finally, rewrite the derivative in a more readable form. The derivative \(f'(x) = -\frac{3}{2}x^{-\frac{5}{2}}\) can also be expressed as \[f'(x) = -\frac{3}{2} \cdot \frac{1}{x^{5/2}}\] or equivalently, \[f'(x) = -\frac{3}{2}\frac{1}{x^{5/2}}\].

Key Concepts

Derivative

The derivative is a fundamental concept in calculus. It measures how a function changes as its input changes.
A derivative represents the rate of change or the slope of the function at any given point:

• In practice, when you find a derivative, you're examining how a small change in the variable affects the function.
• For example, if you have a function describing distance over time, the derivative gives you the speed.

In this problem, we are tasked with finding the derivative of the function \(f(x) = \sqrt{\frac{1}{x^3}}\). This involves determining how this function behaves or how it changes as \(x\) varies. It can be visualized graphically as the slope of the tangent to the curve at any point \(x\). Identifying derivatives is crucial in many real-world applications like optimization and in the analysis of physical systems.

Power Rule

The power rule is a vital tool in differentiation.
This rule significantly simplifies finding derivatives:

• It states that for any function in the form \(f(x) = x^n\), the derivative is \(f'(x) = nx^{n-1}\).
• This means you multiply by the exponent and then decrease the exponent by one.

For example, using the power rule to differentiate \(x^3\), we get \(3x^2\).

Applying the Power Rule

To apply the power rule to our function, first we must express it as \(f(x) = x^{-3/2}\) by rewriting the radical and the fraction using exponents.
Then, the power rule gives us the derivative:

• Multiply the exponent \(-3/2\) by \(x\) and reduce the exponent by 1, resulting in \(-\frac{3}{2}x^{-5/2}\).

This process highlights the power rule's strength by simplifying the differentiation of power functions.

Exponents

Exponents are crucial in manipulating expressions, particularly in calculus.
They indicate how many times a number, known as the base, is multiplied by itself:

• In calculus, understanding exponents allows us to transform functions, making them easier to differentiate.
• This transformation often involves converting roots and fractions to simpler power forms.

For instance, the expression \(\sqrt{\frac{1}{x^3}}\) is easier to tackle when rewritten as \(x^{-3/2}\). Here:

• The square root is represented as \(1/2\) power.
• The reciprocal of \(x^3\) is written as \(x^{-3}\).
• Combining these gives us \(x^{-3/2}\).

By mastering exponents, we can seamlessly manipulate and simplify expressions, paving the way for straightforward differentiation.