Problem 9 - Calculus - AP Edition

Question

$$\text { Find } f^{\prime}(x)$$ $$f(x)=x^{-3}+\frac{1}{x^{7}}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -3x^{-4} - 7x^{-8} \).

Step 1: Express the Function in Terms of Exponents

First, express all terms in the function using exponents. The function given is \( f(x) = x^{-3} + \frac{1}{x^7} \). The term \( \frac{1}{x^7} \) can be rewritten using exponent notation as \( x^{-7} \). Thus, we can rewrite the function as \( f(x) = x^{-3} + x^{-7} \).

Step 2: Differentiate Each Term Individually

Now, differentiate each term of \( f(x) \) with respect to \( x \). For \( x^{-3} \), use the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \). Differentiating \( x^{-3} \) gives \( -3x^{-4} \). Similarly, differentiating \( x^{-7} \) gives \( -7x^{-8} \).

Step 3: Combine Derivatives

Combine the derivatives of each term to find \( f'(x) \). Thus, the derivative is \( f'(x) = -3x^{-4} - 7x^{-8} \).

Step 4: Simplify the Expression

If needed, simplify the expression for better readability. The expression \( f'(x) = -3x^{-4} - 7x^{-8} \) is already simplified, so we can express it as it is or keep in original form if preferred; for example: \( f'(x) = -\frac{3}{x^4} - \frac{7}{x^8} \).

Key Concepts

Power Rule

The Power Rule is a fundamental technique in calculus for differentiating functions involving exponents. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is a real number, you can find the derivative using the formula \( f'(x) = nx^{n-1} \). This means that you multiply the original exponent by the coefficient and then reduce the exponent by one.

When dealing with problems like \( f(x) = x^{-3} + \frac{1}{x^{7}} \), it's crucial to express all terms as exponents. For example, the expression \( \frac{1}{x^7} \) can be rewritten as \( x^{-7} \), which allows the Power Rule to be applied straightforwardly.

• Identify the exponent of each term.
• Apply the Power Rule individually to each term.
• Combine the derivatives to find the result.

Exponents in Calculus

Understanding exponents in calculus is essential for simplifying expressions and differentiating functions. Exponents indicate how many times a number, known as the base, is multiplied by itself. In calculus, exponents can be positive, negative, or even fractions, which all affect how you handle differentiation.

In the function \( f(x) = x^{-3} + x^{-7} \), both terms involve negative exponents. Negative exponents, such as \( x^{-n} \), can be rewritten as \( \frac{1}{x^n} \), providing a way to switch between forms when necessary.

• Negative exponents create terms that are reciprocals.
• The re-expression of fractions as negative exponents is often used before applying differentiation techniques.
• Exponents simplify complex fractions in functions that need to be differentiated.

Derivative Simplification

After finding the derivative using rules like the Power Rule, simplifying the expression is usually the next step. This makes your answer more understandable and ready for further analysis.

For the derivative \( f'(x) = -3x^{-4} - 7x^{-8} \), you can choose to express it in fractions for a different style, like \( f'(x) = -\frac{3}{x^4} - \frac{7}{x^8} \). Simplification doesn't always mean changing the form, but making sure it's clear and accurate.

• Combine like terms if possible.
• Express terms in a preferred readable format.
• Ensure that the derivative maintains its integrity after simplification.

Simplifying derivatives helps in recognizing patterns and making calculus problems easier to handle.