Problem 12 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=7 x^{-6}-5 \sqrt{x} $$

Step-by-Step Solution

Answer

\(f'(x) = -42x^{-7} - \frac{5}{2}x^{-1/2}\)

Step 1: Identify the Function Terms

The given function is \(f(x) = 7x^{-6} - 5\sqrt{x}\). It consists of two terms: the term \(7x^{-6}\) and the term \(-5\sqrt{x}\), which can be rewritten as \(-5x^{1/2}\).

Step 2: Differentiate the First Term

Using the power rule for derivatives, the derivative of \(7x^{-6}\) is found by multiplying the exponent by the coefficient and subtracting one from the exponent. Thus, \((7)(-6)x^{-6-1} = -42x^{-7}\).

Step 3: Differentiate the Second Term

For \(-5x^{1/2}\), use the power rule. Multiply the exponent by the coefficient: \((-5)(1/2)x^{1/2-1} = -(5/2)x^{-1/2}\).

Step 4: Combine the Derivatives

Add the derivatives of the individual terms to find \(f'(x)\): \(-42x^{-7} - \frac{5}{2}x^{-1/2}\).

Key Concepts

Power Rule for Derivatives

The power rule for derivatives is a fundamental tool in calculus that simplifies the differentiation of functions involving terms with exponents. It can be represented as follows: if you have a function of the form \( f(x) = ax^n \), then its derivative \( f'(x) \) can be computed as \( f'(x) = anx^{n-1} \). This rule allows you to quickly calculate how a function's value changes as \( x \) changes.

Here's how it works step by step:

• Identify the term in the form \( ax^n \).
• Multiply the exponent \( n \) by the coefficient \( a \) to find the new coefficient.
• Subtract one from the exponent \( n \) to get the new exponent.

You apply this rule individually to each term in a polynomial or function with multiple terms, as seen in the original exercise. Remember, the power rule simplifies the sometimes complex-looking expressions, making it easier to handle exponents, even negative or fractional ones.

Differentiation Techniques

Differentiation techniques are methods used to find the derivative of a function. Besides the power rule, these include the sum rule, product rule, quotient rule, and chain rule, which can tackle various forms of functions.

In the exercise, we mainly rely on the power rule for simplicity. However, understanding a few additional techniques is beneficial:

• Sum Rule: If you have multiple terms added or subtracted, differentiate them separately and combine their derivatives. This is used in the exercise to differentiate \( 7x^{-6} \) and \(-5x^{1/2} \) individually.
• Product Rule: Useful when differentiating products of two functions, this rule states \((fg)' = f'g + fg'\).
• Quotient Rule: For divisions of functions: \( \left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2} \).
• Chain Rule: Applies for composite functions, expressed as \((f(g(x)))' = f'(g(x))g'(x)\).

By mastering these techniques, you become more versatile in solving diverse mathematical problems, not just those that simply involve the power rule.

Mathematical Notation

Mathematical notation provides a universal language to express mathematical ideas succinctly and clearly. In calculus, this often involves symbols for derivatives and operations. Here are crucial notation concepts used in differentiation:

• Derivatives: Often expressed with \( f'(x) \) for the first derivative of a function \( f(x) \). A prime mark (\(' \)) is a common shorthand for taking the derivative.
• Exponents: Represent how many times a number is multiplied by itself. In differentiation, knowing how to manipulate exponents is key, especially when applying the power rule.
• Radicals as Exponents: Square roots and other roots can be rewritten as fractional exponents, e.g., \( \sqrt{x} = x^{1/2} \). This form is easier to differentiate using the power rule.
• Negative Exponents: Indicate reciprocal powers, so \( x^{-n} = \frac{1}{x^n} \). Handling these correctly is crucial in finding derivatives of terms with negative powers.

Understanding these notations deeply will significantly ease the process of differentiating functions and grasping the underlying calculus concepts.