Problem 56 - Calculus and its applications

Question

Find \(y^{\prime}\) $$ \text { If } y=\frac{1}{3 x^{4}}, \text { find }\left.\frac{d y}{d x}\right|_{x=-1} $$

Step-by-Step Solution

Answer

The derivative at \( x = -1 \) is \( \frac{4}{3} \).

Step 1: Rewrite the Function

First, rewrite the given function for easier differentiation. The function is given as \( y = \frac{1}{3x^4} \), which can be rewritten as \( y = \frac{1}{3}x^{-4} \). This will allow us to apply the power rule when differentiating.

Step 2: Differentiate Using the Power Rule

The power rule for differentiation states that \( \frac{d}{dx}x^n = nx^{n-1} \). Applying this rule to \( y = \frac{1}{3}x^{-4} \), we get: \[ y' = \frac{1}{3}(-4)x^{-5} = -\frac{4}{3}x^{-5} \].

Step 3: Substitute and Simplify

Now, substitute \( x = -1 \) into the derivative to find \( y' \) at \( x = -1 \):\[ y'(-1) = -\frac{4}{3}(-1)^{-5} \].Since \((-1)^{-5} = -1\), this simplifies to:\[ y'(-1) = -\frac{4}{3}(-1) = \frac{4}{3} \].

Key Concepts

The Power Rule

The power rule is a foundational concept in calculus used for differentiating functions. It provides a quick and efficient method to compute the derivative of a power function. Suppose you have a function of the form \( f(x) = x^n \). According to the power rule, the derivative of this function is given by \( \frac{d}{dx}x^n = nx^{n-1} \). This simple formula allows you to transform and find derivatives quickly by bringing down the exponent as a coefficient and reducing the exponent by one.

• This rule is applicable only to functions where the power, \(n\), is a real number.
• It is most frequently used because many expressions in calculus can be simplified to a power rule form.

Understanding the power rule not only accelerates the process of differentiation but helps in approaching more complex calculus concepts with ease.

Derivative Evaluation

Derivative evaluation involves determining the slope or rate of change of a function at specific points. Once you've differentiated a function, the evaluation at a certain point involves substituting the given point into the derived expression. For instance, in our problem, the function was rewritten and differentiated using the power rule. We then substituted the value \( x = -1 \) into the derivative function \( y' = -\frac{4}{3}x^{-5} \).

• Basic rule: Substitute the \(x\)-value into the derivative function.
• Simplify the result to understand the slope of the original function at that point.

After substitution and simplification, you get the value \( \frac{4}{3} \), indicating the slope of the original function at \( x = -1 \). This process translates the instantaneous rate of change or slope at that specific point.

Function Rewriting

Before applying differentiation rules, it often helps to rewrite a function in a more manageable form. For example, if given a function like \( y = \frac{1}{3x^4} \), rewriting it as \( y = \frac{1}{3}x^{-4} \) can make the differentiation process straightforward by aligning it with the power rule.

• Rewriting often involves expressing fractions as negative exponents.
• This step simplifies subsequent calculations and avoids complex algebraic manipulations.

Rewriting functions is a critical skill that helps avoid mistakes and creates clarity in calculus problems. Simplifying expression forms before applying rules is a strategy that pays off when moving to more complex problems.