Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=9 x^{1 / 3}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 9x^{1/3}\) is \(f'(x) = 3x^{-2/3}\).

Step 1: Identify the given function and its components

The function we are given is \(f(x) = 9x^{1/3}\). In this case, our "parent function" is \(x^r\), where r is a real number, and r = 1/3. We also have a constant, 9, multiplied with the parent function.

Step 2: Apply the power rule of differentiation

The power rule of differentiation states that if we have a function \(f(x) = x^r\), then its derivative is given by \(f'(x) = rx^{r-1}\). When we apply this rule to the given function, we have: \(f'(x) = \frac{d}{dx}(9x^{1/3})\) To differentiate this function, we will apply the power rule combined with the constant multiplier: - The constant 9 stays the same - the exponent, 1/3, becomes the coefficient of the term - subtract 1 from the original exponent (1/3 - 1 = -2/3) Following these, we differentiate the given function: \(f'(x) = 9\left(\frac{1}{3}x^{-2/3}\right)\)

Step 3: Simplify the derivative if necessary

Now we will simplify the derivative obtained in step 2: \(f'(x) = 9\left(\frac{1}{3}x^{-2/3}\right) = 3x^{-2/3}\)

Step 4: Write down the final result

We have successfully found the derivative of the given function using the power rule. So, the final result is: \(f'(x) = 3x^{-2/3}\)

Key Concepts

Power Rule in Differentiation

The power rule is a fundamental concept in calculus used to find the derivative of a function. It's especially handy when dealing with polynomials or any function of the form \(x^r\).
The rule states that if you have a function \(f(x) = x^r\), then its derivative \(f'(x)\) is calculated as:

• Multiply the exponent \(r\) by the base \(x\).
• Reduce the original exponent by 1.
• This gives the new exponent of \(x\) in the derivative.

In our example, we used the function \(f(x) = 9x^{1/3}\). By applying the power rule, the derivative becomes \(f'(x) = \frac{1}{3}x^{-2/3}\). The factor 9 remains because it is a constant multiplier. Differentiation.rules often keep constants unchanged.

Understanding the Derivative of a Function

The derivative of a function is a cornerstone concept in calculus. It represents the rate at which a function is changing at any given point.
When you differentiate a function, you're essentially seeing how much the function value changes as the input changes a little bit.
For the exercise \(f(x) = 9x^{1/3}\), finding the derivative means identifying how \(f(x)\) changes as \(x\) changes.

• The derivative, denoted \(f'(x)\), is found using the power rule.
• It lets us determine the slope of the tangent line to the curve at any point \(x\).

Ultimately, differentiation provides essential information about the behavior of functions so that we can better analyze them.

Basics of Calculus and Differentiation

Calculus is a branch of mathematics that studies how things change. At its core are two primary operations: differentiation and integration.
Differentiation deals with finding how a function changes — its derivatives. This is important because it lets us understand growth, decay, velocity, or any type of change.

• Calculus is used extensively in science, economics, biology, engineering, and many fields requiring modeling and change prediction.
• Differentiation involves various rules, including the power rule, product rule, and chain rule.

Getting a grip on derivatives through calculus requires practice and understanding each rule's application.
A foundation in calculus equips you to tackle complex functions and their changes, vital for various types of analytical work.