Problem 35 - Calculus and its applications

Question

Find \(f^{\prime}(x)\). $$ f(x)=\frac{2}{5 x^{6}} $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{12}{5x^7}\).

Step 1: Identify the Function Type

The given function is \[ f(x) = \frac{2}{5x^6} \]This can be rewritten as a power of \(x\) by expressing the denominator as a negative exponent. Thus, we have:\[ f(x) = \frac{2}{5}x^{-6} \]

Step 2: Apply the Power Rule for Differentiation

The power rule for differentiation states that if \( f(x) = ax^n \), then the derivative \( f'(x) = a\cdot n\cdot x^{n-1} \). For this function, \( a = \frac{2}{5} \) and \( n = -6 \). Therefore, we apply the power rule:\[ f'(x) = \frac{2}{5}(-6)x^{-6-1} \]

Step 3: Simplify the Derivative Expression

Calculating further, we get:\[ f'(x) = \frac{2}{5} \times (-6) x^{-7} \]This simplifies to:\[ f'(x) = -\frac{12}{5} x^{-7} \]

Step 4: Rewrite with Positive Exponents

If desired, you can rewrite the expression back in terms of positive exponents:\[ f'(x) = -\frac{12}{5} \cdot \frac{1}{x^7} \]This simplifies to:\[ f'(x) = -\frac{12}{5x^7} \]

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus that simplifies the process of differentiation. Differentiation is the calculation of a derivative, which measures how a function changes as its input changes. The Power Rule is applied to functions of the form \(f(x) = ax^n\), where \(a\) and \(n\) are constants. The rule states: bring down the exponent \(n\) in front as a coefficient and then subtract one from the exponent.

• If \(f(x) = ax^n\), then \(f'(x) = a \cdot n \cdot x^{n-1}\).
• This rule simplifies the differentiation process by providing a quick method to find derivatives of power functions.

Using the Power Rule can make solving calculus problems faster, especially those involving polynomial functions. In our example, rewriting the given function \(f(x) = \frac{2}{5} x^{-6}\), the power rule is applied directly to simplify and solve for its derivative.

Negative Exponents

Negative exponents can initially seem confusing, but they are actually a straightforward concept once understood. A negative exponent indicates that instead of multiplying, you are taking the reciprocal of the base raised to the corresponding positive exponent.

• For example, the expression \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\).
• This means \(x^{-6}\) can be written as \(\frac{1}{x^6}\).

In our exercise, the original function is given with \(x^6\) in the denominator: \(f(x) = \frac{2}{5x^6}\). By expressing this with a negative exponent, \(f(x) = \frac{2}{5}x^{-6}\), it becomes easier to apply the power rule, as the exponent is now manageable for differentiation.
Using negative exponents helps to simplify expressions before applying calculus rules like differentiation.

Derivative Calculation

Calculating a derivative is about finding the rate at which a function is changing at any given point. For functions involving powers, this task is simplified by using the Power Rule, with careful attention to handling exponents.
Here is how we applied derivative calculation in today's example:

• We started with the function \(f(x) = \frac{2}{5}x^{-6}\).
• The power rule was applied: multiply the coefficient by the exponent and decrease the exponent by one to find \(f'(x)\).
• This led to: \(f'(x) = \frac{2}{5} \times (-6)x^{-7}\).
• After simplifying, the derivative \(f'(x)\) becomes \(-\frac{12}{5}x^{-7}\).
• If needed, rewrite using positive exponents: \(-\frac{12}{5}\cdot\frac{1}{x^7} = -\frac{12}{5x^7}\).

Derivative calculations allow us to understand how functions behave and change, and mastering this skill is crucial in calculus for analyzing functions' behaviors efficiently.