Problem 108 - Calculus and its applications

Question

Find dy/dx. Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$ y=\sqrt[3]{8 x} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{2}{3x^{2/3}} \).

Step 1: Rewriting the Expression

First, rewrite the given function using fractional exponents. The expression \( y = \sqrt[3]{8x} \) can be rewritten as \( y = (8x)^{1/3} \). This will make the process of differentiation simpler.

Step 2: Apply the Chain Rule

Consider \( u = 8x \), thus \( y = u^{1/3} \). By the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). We need to differentiate \( u^{1/3} \) and \( 8x \) with respect to \( u \) and \( x \), respectively.

Step 3: Differentiate with respect to u

Differentiate \( y = u^{1/3} \) with respect to \( u \). The derivative is \( \frac{dy}{du} = \frac{1}{3}u^{-2/3} \).

Step 4: Differentiate u with respect to x

Differentiate \( u = 8x \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 8 \).

Step 5: Combine the results

Substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) back into the chain rule equation: \( \frac{dy}{dx} = \frac{1}{3}u^{-2/3} \cdot 8 \).

Step 6: Substitute back u in terms of x

Replace \( u \) with \( 8x \) to express the derivative in terms of \( x \): \[ \frac{dy}{dx} = \frac{8}{3}(8x)^{-2/3} \].

Step 7: Simplify the Expression

Simplify the expression for \( \frac{dy}{dx} \) as needed. The final derivative is \[ \frac{dy}{dx} = \frac{8}{3} \cdot \frac{1}{(8x)^{2/3}} = \frac{8}{3} \cdot \frac{1}{4x^{2/3}} = \frac{2}{3} \cdot \frac{1}{x^{2/3}} = \frac{2}{3x^{2/3}} \].

Key Concepts

Chain Rule

The Chain Rule is a method in calculus for differentiating composite functions. Composite functions are those where a function is nested within another function. If you have a function of the form \( f(g(x)) \), the Chain Rule helps you find the derivative by first differentiating the outer function while keeping the inner function the same. Then you multiply the result by the derivative of the inner function. This might sound confusing at first, but here's a simple breakdown:

• Differentiate the outer function: If \( y = u^{1/3} \), find \( \frac{dy}{du} = \frac{1}{3}u^{-2/3} \).
• Differentiate the inner function: If \( u = 8x \), find \( \frac{du}{dx} = 8 \).
• Multiply these derivatives: \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \).

The result is a new function representing the rate of change of \( y \) with respect to \( x \).

Fractional Exponents

Fractional exponents are an alternative way to express roots. For example, a cube root such as \( \sqrt[3]{8x} \) can be rewritten using fractional exponents as \( (8x)^{1/3} \). This form is particularly useful in calculus because it simplifies the process of differentiation.

When using fractional exponents, remember these basic rules:

• If \( a^{m/n} \), it means \( \sqrt[n]{a^m} \).
• The rules of exponentiation still apply, such as \( a^{m} \cdot a^{n} = a^{m+n} \), and \( (a^{m})^{n} = a^{m \cdot n} \).
• Fractional exponents help us apply simpler differentiation rules, as seen in the Chain Rule application.

Using fractional exponents can make complex calculus operations more manageable and readable.

Algebraic Manipulation

Algebraic Manipulation involves rearranging expressions to simplify calculations or solve equations more easily. In differentiation, this often includes rewriting expressions before applying calculus rules.

Take our original function \( y = \sqrt[3]{8x} \), which is easier to differentiate after being rewritten as \( y = (8x)^{1/3} \). By rewriting, the power rule can be applied directly.

Key steps in algebraic manipulation include:

• Identifying forms like roots or complex fractions that can simplify through rewriting.
• Applying exponent rules to rewrite expressions in terms of simpler exponents.
• Simplifying expressions post-differentiation to make derivatives easier to understand and use.

Simple algebraic manipulations make complex differentiation processes straightforward and efficient.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often comes into play after differentiation to make the result cleaner and easier to understand.

In our solution, we stepped through simplifying the derivative from \( \frac{dy}{dx} = \frac{8}{3}(8x)^{-2/3} \) to a more intuitive form:

• Naturally cancelling terms if possible.
• Applying basic algebra to combine like terms and reduce factors.
• Continuing simplification until you have the simplest and most readable form: \( \frac{2}{3x^{2/3}} \).

By simplifying expressions, we ensure that the final solution is not just correct, but also transparent and ready for application in further mathematical analysis.