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Chapter 2: Problem 15

Find the second derivative of each function. $$f(x)=\frac{27}{\sqrt[3]{x}}$$

Short Answer

Expert verified
The second derivative is \( f''(x) = 12x^{-7/3} \).

Step by step solution

01

Rewrite the function

The given function is \( f(x) = \frac{27}{\sqrt[3]{x}} \). We can rewrite this in terms of exponents to make differentiation easier. Remember that \( \sqrt[3]{x} \) is the same as \( x^{1/3} \). So, \( f(x) = 27x^{-1/3} \).
02

Find the first derivative

To find the first derivative, apply the power rule for differentiation, which is \( \frac{d}{dx}[x^n] = nx^{n-1} \). Thus, for \( f(x) = 27x^{-1/3} \), we have:\( f'(x) = 27 \cdot (-1/3)x^{-1/3 - 1} \), which simplifies to \( f'(x) = -9x^{-4/3} \).
03

Find the second derivative

To find the second derivative, differentiate \( f'(x) = -9x^{-4/3} \) again using the power rule. The derivative is:\( f''(x) = -9 \cdot (-4/3)x^{-4/3 - 1} \), which simplifies to \( f''(x) = 12x^{-7/3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule for Differentiation
The power rule is a fundamental tool in calculus for finding the derivative of a function in the form of \( x^n \). It's simple yet powerful. When you want to differentiate \( x^n \), the power rule tells you to multiply the power \( n \) by the coefficient of \( x \), and then subtract one from the exponent. For example:
  • If \( f(x) = x^3 \), then \( f'(x) = 3x^{2} \).
Applying this to more complex functions, like the ones in our exercise, we can differentiate terms like \( 27x^{-1/3} \).
  • First, select the exponent: here \( n = -1/3 \).
  • Then multiply the coefficient 27 by \(-1/3\) and subtract one from the exponent, resulting in \( f'(x) = -9x^{-4/3} \).
This simplicity makes the power rule a favorite for differentiating polynomials and similar expressions.
Exponents in Calculus
Exponents are a critical part of calculus, and they're everywhere in mathematical expressions. Understanding how to manipulate and differentiate expressions with exponents allows you to solve many calculus problems efficiently. In the exercise, we had \( f(x) = \frac{27}{\sqrt[3]{x}} \), which required rewriting into an exponent form \( f(x) = 27x^{-1/3} \).
  • This allows us to apply differentiation rules effectively. Negative and fractional exponents represent roots and reciprocals:
  • \( x^{-1/3} \) becomes \( 1/x^{1/3} \), the cube root being represented as \( x^{1/3} \).
These conversions are essential when preparing expressions for differentiation. Always remember, you can rewrite roots as fractional exponents to simplify calculus operations.
Differentiation Techniques
Finding derivatives is a key part of calculus, and using the right differentiation techniques makes this process smoother. Differential calculus primarily deals with the rate of change, and knowing how to handle these changes leads to finding the derivative efficiently. In the example, after rewriting the function, the next step was to differentiate it twice to get the second derivative.
  • Applying the power rule: First, derive \( 27x^{-1/3} \) to get \( f'(x) = -9x^{-4/3} \).
  • Then, apply the rule again to \( f'(x) \), simplifying to get \( f''(x) = 12x^{-7/3} \).
Using differentiation techniques like these ensures accurate results and deeper understanding. These practices underline the importance of knowing how to handle each step methodically.

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Most popular questions from this chapter

Suppose that \(E(x)\) is a function such that \(E^{\prime}(x)=E(x)\). Use the Chain Rule to show that the derivative of the composite function \(E(g(x))\) is \(\frac{d}{d x} E(g(x))=E(g(x)) \cdot g^{\prime}(x)\).

Use the Generalized Power Rule to find the derivative of each function. $$f(x)=\frac{1}{\sqrt[3]{\left(2 x^{2}-3 x+1\right)^{2}}}$$

Find the number \(x\) of units at which the marginal cost is \(1.75\). [Hint: TRACE along the marginal cost function \(1 / 2\) to find where the \(y\) -coordinate is \(1.75\), giving your answer as the \(x\) -coordinate rounded to the nearest whole number.]

Use the Generalized Power Rule to find the derivative of each function. $$f(x)=x^{2} \sqrt{x^{2}-1}$$

After \(p\) practice sessions, a subject could perform a task in \(T(p)=36(p+1)^{-1 / 3}\) minutes for \(0 \leq p \leq 10\) Find \(T^{\prime}(7)\) and interpret your answer.
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