Problem 7 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=\frac{1}{x^{5}} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=\frac{1}{x^{5}}\) is \(y' = -\frac{5}{x^{6}}\)

Step 1: Rewrite the function

The function \(y=\frac{1}{x^{5}}\) can be rewritten as \(y = x^{-5}\). This form allows us to easily apply the power rule for differentiation. In this case the power is -5.

Step 2: Apply the power rule

Applying the power rule for differentiation, we get \(y' = -5*x^{-5-1}\) which simplifies to \(y' = -5*x^{-6}\).

Step 3: Simplify the result

The simplified expression of the derivative is \(y' = -\frac{5}{x^{6}}\)

Key Concepts

Power Rule

The power rule is one of the foundational techniques used in calculus for finding the derivative of a function with a simple exponent. It states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, the derivative of this function, denoted as \( f'(x) \) or \( \frac{df}{dx} \), is given by multiplying the exponent with the variable raised to one power less than the original. That is, \( f'(x) = nx^{n-1} \).

In the context of the exercise provided, we started with the function \( y = \frac{1}{x^5} \), which can also be written as \( y = x^{-5} \). By applying the power rule, we find the derivative by multiplying the exponent, -5, by the variable \( x \) raised to the new power, which is one less than -5, yielding \( y' = -5x^{-6} \). This simple, yet powerful rule, allows for quick differentiation without the need for complex calculations or limits.

Derivative of a Function

The derivative of a function represents the rate at which the function's output changes with respect to changes in the input. In simpler terms, it measures the slope or steepness of the curve at any particular point. Derivatives are fundamental in calculus and are used to solve problems involving rates of change, such as acceleration in physics or marginal cost in economics.

To find the derivative of a function, we apply various differentiation rules such as the power rule, product rule, quotient rule, or chain rule, depending on the structure of the function. For example, in the exercise \( y = \frac{1}{x^5} \), we are dealing with a power of \( x \), therefore we utilize the power rule for differentiation to efficiently find the derivative.

Simplifying Derivatives

Simplifying derivatives is a crucial step in making the results of differentiation more comprehensible and easier to use in further mathematical or real-world applications. Simplification might involve rewriting negative exponents as positive ones in a fraction, combining like terms, or factoring out common factors.

In the exercise, after applying the power rule, we obtained \( y' = -5x^{-6} \). To simplify this derivative, we can express the negative exponent as a positive one in the denominator of a fraction, resulting in \( y' = -\frac{5}{x^6} \). This final form clearly reveals the nature of the relationship between the variables, emphasizing the inverse nature of the derivative's magnitude with respect to the input variable.