Problem 18 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{4}{\sqrt{x}} $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Function

First, rewrite the function in a form that is easier to differentiate using the power rule. The given function is \( f(x) = \frac{4}{\sqrt{x}} \). We can express the square root as a power: \( \sqrt{x} = x^{1/2} \). Therefore, the function becomes \( f(x) = 4x^{-1/2} \).

Step 2: Apply the Power Rule

Now, apply the power rule for differentiation, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Here, \( n = -1/2 \) and the function is \( f(x) = 4x^{-1/2} \). Differentiating, we get:\[ f'(x) = 4 \cdot (-\frac{1}{2})x^{-3/2} = -2x^{-3/2} \].

Step 3: Simplify the Result

Express \( f'(x) \) in a simpler form by rewriting \( x^{-3/2} \) as a fraction. We have:\[ f'(x) = -\frac{2}{x^{3/2}} \]. This rewritten form is often preferred as it avoids negative exponents.

Key Concepts

Understanding the Power Rule

The power rule is a quick and effective technique used to find the derivative of a function of the form \( f(x)=x^n \). This rule is essential in calculus and simplifies the differentiation process significantly. To apply the power rule, you simply need to:

• Multiply the original exponent \( n \) by the coefficient in front of \( x \).
• Reduce the exponent by one.

For instance, if you have a term \( 4x^{-1/2} \), the power rule allows you to find its derivative by performing the following steps: multiply \(-1/2\) by 4, resulting in \(-2\), and then subtract one from \(-1/2\), giving a new exponent of \(-3/2\). This gives the derivative of the term as \(-2x^{-3/2}\). The simplicity of the power rule comes from its consistent application, making it an invaluable tool in calculus.

Basics of Function Differentiation

Differentiation is a cornerstone of calculus, involving the calculation of a function's derivative. The derivative represents the rate at which a function is changing at any given point. For basic algebraic functions, such as polynomials, differentiation is quite straightforward. You can differentiate each term individually and then combine the results. When handling functions like \( f(x)=\frac{4}{\sqrt{x}} \), change the expression to make it easier for differentiation.

• Rewrite \( \sqrt{x} \) as \( x^{1/2} \).
• Convert the fraction \( \frac{4}{x^{1/2}} \) into \( 4x^{-1/2} \).

This transformation makes it simpler to apply the power rule. Differentiation is crucial because it helps understand how a function behaves and predict changes under different conditions.

Simplifying Expressions

After finding the derivative, simplifying the expression can lead to a cleaner, more easily interpretable form. This final step in differentiation involves expressing derivatives with negative exponents into fractions, making them more palatable. In the case of \( f'(x) = -2x^{-3/2} \), convert it as:

• Rewrite \( x^{-3/2} \) as \( \frac{1}{x^{3/2}} \).
• The expression becomes \( f'(x) = -\frac{2}{x^{3/2}} \).

Simplified expressions often provide a clearer picture of the derivative's behavior and make further calculations more manageable. This step ensures the derivative maintains the same properties as the original function, just in a tidier format that avoids the potential confusion of negative exponents.