Question

Finding a Second Derivative In Exercises \(91-98\) , find the second derivative of the function. $$ f(x)=4 x^{3 / 2} $$

Short answer

The second derivative of the function \(f(x)=4*x^{3/2}\) is \(f''(x)=3*x^{-1/2}\)

Step-by-step solution

Finding the First Derivative

The first derivative of a function \(f(x)\) can be found using the power rule, which is stated as \((f(x)=x^n)'=n*x^{n-1}\). So for the given function \(f(x)=4x^{3/2}\), the first derivative is \((4*x^{3/2})'=3/2*4*x^{1/2}=6*x^{1/2}\).

Finding the Second Derivative

The second derivative of a function \(f(x)\) can be found by taking the derivative of the first derivative. Therefore, to find the second derivative of the given function, you should find the derivative of \(f'(x)=6*x^{1/2}\). Applying the power rule again, you get \((6*x^{1/2})'=1/2*6*x^{-1/2}=3*x^{-1/2}\).

Key concepts

Power Rule

The power rule is a fundamental tool in calculus that simplifies the process of differentiation.
It calculates derivatives of functions in the form of power functions. Specifically, the power rule states that if you have a function of the form \( f(x) = x^n \), then the derivative \( f'(x) \) is \( nx^{n-1} \).
This rule allows us to easily differentiate functions by handling each term separately.

• If you have a function like \( f(x) = 4x^{3/2} \), each term is differentiated using the power rule individually.
• The coefficient, in this case 4, is multiplied by the exponent \( n \), which is \( 3/2 \) here.
• Then, reduce the exponent by one, changing \( 3/2 \) to \( 1/2 \).

This results in the differentiated term \( 6x^{1/2} \), making it easy to proceed with finding higher-order derivatives.

First Derivative

The first derivative of a function represents the rate of change or the slope of the function at any given point.
Using the power rule, the first derivative of \( f(x) = 4x^{3/2} \) is found by multiplying the original exponent by the coefficient, giving \( 6x^{1/2} \).

• The first derivative gives us insight into the function's behavior, such as where it is increasing or decreasing.
• For example, a positive first derivative indicates an increasing function, while a negative first derivative shows a decrease.

Understanding the first derivative is crucial as it provides the foundation for exploring the curvature and concavity of the graph through the second derivative.

Derivative of Power Functions

When dealing with power functions, calculating derivatives helps us understand how they change.
A power function is any function of the form \( f(x) = ax^n \).
Given the derivative \( f'(x) \) from step 1, the second derivative \( f''(x) \) is found by applying the power rule to \( 6x^{1/2} \).

• The second derivative \( f''(x) = 3x^{-1/2} \) gives a deeper insight, showing how the rate of change is itself changing.
• This is particularly useful for understanding the shape and inflection points of the graph of the function.

The process of finding the second derivative from a power function involves repeating the power rule, emphasizing how each differentiation process simplifies the function to explore its characteristics further.