Problem 27 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{10}{\sqrt{x}}-\frac{9 \sqrt[3]{x^{5}}}{5}+17 $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -5x^{-3/2} - 3x^{2/3} \).

Step 1: Rewrite the function

Begin by expressing the function in a form that is easier to differentiate using power rules. Rewrite each term using exponents instead of roots. The function \( f(x) = \frac{10}{\sqrt{x}} - \frac{9 \sqrt[3]{x^{5}}}{5} + 17 \) can be rewritten as \( f(x) = 10x^{-1/2} - \frac{9}{5}x^{5/3} + 17 \).

Step 2: Differentiate the first term

Apply the power rule to differentiate the first term \(10x^{-1/2}\). The power rule \( \frac{d}{dx} [x^n] = nx^{n-1} \) gives us \( \frac{d}{dx}[10x^{-1/2}] = 10 \cdot (-1/2)x^{-1/2-1} = -5x^{-3/2} \).

Step 3: Differentiate the second term

Now differentiate the second term \(-\frac{9}{5}x^{5/3}\) using the power rule. This gives \( \frac{d}{dx}[-\frac{9}{5}x^{5/3}] = -\frac{9}{5} \cdot \frac{5}{3}x^{5/3-1} = -3x^{2/3} \).

Step 4: Differentiate the constant term

Since the derivative of a constant is zero, the derivative of the constant 17 is simply \(0\).

Step 5: Combine derivatives

Add the derivatives of each term to find the derivative of the entire function. Thus, the derivative \( f'(x) \) is \(-5x^{-3/2} - 3x^{2/3} + 0 \).

Step 6: Simplify the derivative

Combine the terms to simplify the derivative expression: \( f'(x) = -5x^{-3/2} - 3x^{2/3} \). This is the final derivative.

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus used to find derivatives of polynomial expressions. It's especially useful when dealing with terms involving variables raised to constant powers.
When applying the power rule, the exponent of the term is multiplied by the coefficient, then the exponent is reduced by one. This simple method allows for quick differentiation, making it a handy tool for math students.

The general formula is:

• If you have a term in the form \( x^n \), the derivative is given by \( \frac{d}{dx} [x^n] = nx^{n-1} \).
• Notice how the exponent \( n \) is brought to the front as a multiplier, and the original power is decreased by one.

To illustrate, consider the function \( 10x^{-1/2} \). Using the power rule:
- The exponent \( -1/2 \) is multiplied by 10, yielding \( 10(-1/2) = -5 \).
- Then, the exponent is decreased by 1, changing from \( -1/2 \) to \( -3/2 \), giving \( -5x^{-3/2} \).
Mastering the power rule helps in efficiently tackling derivatives of polynomial functions.

Differentiation

Differentiation is the mathematical process of finding the derivative of a function. The derivative represents the rate at which a function changes as its input changes. In simpler terms, it's a way to figure out how a function is behaving at any given point.
By differentiating a function, we can determine its slope, make predictions about its growth, and analyze its features such as maxima and minima.

The procedure of differentiation involves:

• Identifying the function you wish to differentiate.
• Choosing the appropriate rule or method. Common methods include the power rule or chain rule, among others.
• Applying the rule to find the derivative.

For the function \( f(x) = 10x^{-1/2} - \frac{9}{5}x^{5/3} + 17 \), the differentiation process focuses on each term individually. For instance:

• The term \( 10x^{-1/2} \) differentiates to \( -5x^{-3/2} \) using the power rule.
• The second term, \( -\frac{9}{5}x^{5/3} \), becomes \( -3x^{2/3} \).
• The constant 17 turns into 0, as the derivative of any constant is zero.

Once these derivatives are computed, they are combined to form the derivative for the whole function: \( f'(x) = -5x^{-3/2} - 3x^{2/3} \). Understanding these steps in differentiation aids in grasping how changes in a variable affect the function's value.

Exponent Rules

Exponent rules are essential when working with derivatives of functions involving powers or roots. These rules help convert terms into a format that is easier to differentiate.
When differentiating, simplifying a function using exponent rules is often crucial. For example:

Key rules include:

• To rewrite roots as exponents: \( x^{1/n} \) is the n-th root of x.
• For a fraction \( \frac{a}{b}\), write it as \( a \times b^{-1} \) to ease calculus operations.

Take the function \( \frac{10}{\sqrt{x}} - \frac{9 \sqrt[3]{x^{5}}}{5} + 17 \):

• \( \frac{10}{\sqrt{x}} \) is rewritten as \( 10x^{-1/2} \) because \( \sqrt{x} = x^{1/2} \).
• The term \( \sqrt[3]{x^{5}} \) becomes \( x^{5/3} \), allowing easy application of the power rule.

Using these transformations, the function is ready for differentiation. Simplifying terms with exponent rules makes the derivative calculation straightforward and reduces the chances of errors. By becoming familiar with exponent rules, students can simplify complex expressions and differentiate them more easily.