Problem 26 - Brief Applied Calculus

Question

Evaluate each expression. $$ \left.\frac{d^{2}}{d x^{2}} \sqrt[3]{x^{4}}\right|_{x=1 / 27} $$

Step-by-Step Solution

Answer

The value is 4.

Step 1: Simplify the Expression

First, simplify the expression \( \sqrt[3]{x^4} \) to a form that is easier to differentiate. This can be rewritten using exponent rules as \( x^{4/3} \).

Step 2: Calculate the First Derivative

Differentiate \( x^{4/3} \) with respect to \( x \) using the power rule. The derivative of \( x^n \) is \( nx^{n-1} \). So, the first derivative is \( \frac{4}{3}x^{1/3} \).

Step 3: Calculate the Second Derivative

Differentiate \( \frac{4}{3}x^{1/3} \) again with respect to \( x \). The derivative is \( \frac{4}{9}x^{-2/3} \).

Step 4: Evaluate the Second Derivative at \( x=1/27 \)

Substitute \( x = 1/27 \) into the second derivative \( \frac{4}{9}x^{-2/3} \). Simplify to find: \( \frac{4}{9} \left( \frac{1}{27} \right)^{-2/3} = \frac{4}{9} \times 9 = 4 \), since \( \left( 27^{1/3} \right)^{-2} = 9 \).

Key Concepts

Power Rule

The Power Rule is one of the most essential tools when you first learn calculus. It's a shortcut to finding derivatives of functions of the form \( x^n \). Here's how it works:

• Identify the power \( n \) in the expression \( x^n \).
• Bring the exponent \( n \) down in front as a coefficient.
• Reduce the original exponent by 1 to get the new exponent \( n-1 \).

So, the derivative of \( x^n \) is \( nx^{n-1} \). In our exercise, we have a function \( x^{4/3} \). Applying the Power Rule:

• We bring down \( \frac{4}{3} \) as a coefficient.
• Subtract \( 1 \) from the exponent, \( \frac{4}{3} - 1 = \frac{1}{3} \).

Thus, the derivative of \( x^{4/3} \) is \( \frac{4}{3}x^{1/3} \). This simple process allows us to move quickly through complex differentiation problems!
Remember, practicing this rule is key to mastering calculus derivatives.

Second Derivative

Once you've found the first derivative, you may need to find the second derivative. The second derivative tells us about the concavity of the function.In our specific problem, after finding the first derivative \( \frac{4}{3}x^{1/3} \), we need to differentiate this again for the second derivative. Just like before, we use the Power Rule:

• Bring down the coefficient from the first derivative which is \( \frac{4}{3} \).
• The exponent on \( x \) is \( \frac{1}{3} \), so when reduced by one power, it becomes \( -\frac{2}{3} \).

Hence, the second derivative of \( x^{4/3} \) becomes \( \frac{4}{9}x^{-2/3} \). This second derivative indicates whether the function is concave up or down at any point.For our exercise, evaluating this at \( x=1/27 \) lets us see specifically the rate of change of the slope of the tangent to the curve at that point.

Exponent Rules

Exponent rules are powerful tools that simplify expressions and make differentiation easier. In the problem, we start with \( \sqrt[3]{x^4} \). Using exponent rules, we can rewrite this as \( x^{4/3} \).Key exponent rules include:

• For radicals: \( \sqrt[n]{x^m} = x^{m/n} \).
• When multiplying like bases: \( x^a \times x^b = x^{a+b} \).
• When raising a power to a power: \( (x^a)^b = x^{a \times b} \).
• When dividing like bases: \( x^a \div x^b = x^{a-b} \).

In our example, using these rules to write \( \sqrt[3]{x^4} \) as \( x^{4/3} \) is crucial. It lets us apply the Power Rule for differentiation. Understanding and applying exponent rules correctly streamlines your calculations in calculus.