Problem 7 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ F(x)=x^{2} \sqrt{x}-5 x $$

Step-by-Step Solution

Answer

First derivative: \(\frac{5}{2} x^{3/2} - 5\). Second derivative: \(\frac{15}{4} x^{1/2}\)

Step 1: Write down the function

The given function is \[ F(x) = x^2 \sqrt{x} - 5x \]

Step 2: Simplify the function

Rewrite the function to make differentiation easier. Note that \sqrt{x} = x^{1/2}. \[ F(x) = x^2 \cdot x^{1/2} - 5x = x^{5/2} - 5x \]

Step 3: Find the first derivative

Differentiate each term of the simplified function using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \). \[ F^{'}(x) = \frac{d}{dx}(x^{5/2}) - \frac{d}{dx}(5x) \ = \frac{5}{2} x^{3/2} - 5 \]

Step 4: Find the second derivative

Differentiate the first derivative. \[ F^{''}(x) = \frac{d}{dx} \[ \frac{5}{2} x^{3/2} - 5 \] = \frac{5}{2} \cdot \frac{3}{2} x^{1/2} = \frac{15}{4} x^{1/2} \]

Key Concepts

First Derivative

In calculus, the first derivative of a function measures the rate at which the function's value changes with respect to a change in the input value. Finding the first derivative is akin to finding the slope of the tangent line to the function at any given point.
Generally, the notation for the first derivative of a function \( f(x) \) is \( f'(x) \) or \( \frac{d}{dx} f(x) \). The process involves differentiating each term of the function individually and applying the power rule wherever applicable.

Second Derivative

The second derivative provides information on the curvature or concavity of the original function. It tells us how the first derivative changes as the input value changes. A positive second derivative indicates that the original function is concave up, while a negative second derivative indicates concave down.
Similar to the first derivative, finding the second derivative involves differentiating the first derivative.

Power Rule

The power rule is one of the most basic and essential rules in differentiation. It states that if you have a function of the form \( x^n \), the derivative of this function is \( nx^{n-1} \).
Using the power rule simplifies the differentiation process significantly, making it easier to compute the derivatives of polynomial functions.

• For example, if we have \( x^3 \), the derivative is \( 3x^2 \).
• For another term like \( 5x \), it becomes 5...