Problem 25 - Precalculus: Mathematics for Calculus

Question

Find the derivative of the function at the given number. $$F(x)=\frac{1}{\sqrt{x}}, \quad \text { at } 4$$

Step-by-Step Solution

Answer

The derivative at \( x = 4 \) is \( F'(4) = -\frac{1}{16} \).

Step 1: Rewrite the Function

First, let's rewrite the function in a form that is easier to differentiate. The given function is \( F(x) = \frac{1}{\sqrt{x}} \). We can express this as a power of \( x \): \( F(x) = x^{-1/2} \).

Step 2: Differentiate the Function

Now, differentiate \( F(x) = x^{-1/2} \) using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Applying the power rule:\[ F'(x) = -\frac{1}{2} x^{-3/2} \]

Step 3: Evaluate the Derivative at the Given Number

Substitute \( x = 4 \) into the derivative \( F'(x) = -\frac{1}{2} x^{-3/2} \):\[ F'(4) = -\frac{1}{2} \times 4^{-3/2} \]Simplify the expression:First, calculate \( 4^{-3/2} \).Write 4 as \( 2^2 \), so \( 4^{-3/2} = (2^2)^{-3/2} = 2^{-3} = \frac{1}{8} \).Therefore, \( F'(4) = -\frac{1}{2} \times \frac{1}{8} = -\frac{1}{16} \).

Key Concepts

power rule

The power rule is a fundamental tool in calculus used to find the derivative of a function. It's particularly useful when dealing with polynomial expressions. The power rule formula is simple and easy to memorize: if you have a function of the form \[ f(x) = x^n \] then its derivative is given by \[ f'(x) = nx^{n-1} \].
This rule is crucial because it allows us to quickly find the slope of a tangent line to the curve of a function. For example, consider when we need to differentiate \( F(x) = x^{-1/2} \). Applying the power rule here means multiplying the exponent (-1/2) by \( x^{-1/2-1} \), resulting in \( F'(x) = -\frac{1}{2}x^{-3/2} \).

The power rule saves time and effort, replacing what could be a long and complex process of differentiation with a straightforward calculation.

differentiation

Differentiation is a core concept in calculus involving the calculation of a function's derivative. The derivative represents an instantaneous rate of change and is essential for understanding how functions behave.
The process of differentiation provides a way to calculate the slope of the tangent line at any point on a curve. It's the procedure by which a function is broken down to find its derivative, often using various rules like the power rule.

In our example, we started with \( F(x) = \frac{1}{\sqrt{x}} \) and converted it to \( x^{-1/2} \) for easier differentiation. By applying the power rule, we differentiated \( F(x) \) to find that \( F'(x) = -\frac{1}{2}x^{-3/2} \), which shows how \( F(x) \) changes at any given point.
Differentiation is essential not just in mathematics but in fields such as physics and engineering, where understanding change is crucial.

calculus

Calculus is a branch of mathematics focused on studying change. Its main pillars are differentiation and integration. Calculus provides the tools to describe and analyze rates of change and areas under curves.

It was invented to solve problems in physics, astronomy, and other sciences, where understanding change over time and space is necessary. Differentiation, one of the core components of calculus, allows us to find the derivative of functions, as demonstrated by the function \( F(x) = \frac{1}{\sqrt{x}} \). By finding the derivative \( F'(x) \), we can comprehend how \( F \) defines a rate of change in its graph.
Understanding the basics of calculus, like limits, derivatives, and integrals, gives you immense power to tackle complex real-world problems, from predicting the trajectory of planets to designing sophisticated engineering systems.