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Chapter 1: Problem 52

Find the indicated derivative. $$\frac{d}{d x}\left(x^{-1 / 3}\right)$$

Short Answer

Expert verified
\( \frac{d}{dx} x^{-1 / 3} = -\frac{1}{3} x^{-4 / 3} \)

Step by step solution

01

Identify the function

The function given is \[ f(x) = x^{-1 / 3} \].
02

Use the power rule for derivatives

Recall that the power rule states \[ \frac{d}{dx} x^n = nx^{n-1} \]. Here, the exponent is \( n = -\frac{1}{3} \).
03

Apply the power rule

Using the power rule, take the derivative: \[ \frac{d}{dx} x^{-1 / 3} = -\frac{1}{3} x^{-1 / 3 - 1} \].
04

Simplify the expression

Combine the exponents: \[ x^{-1 / 3 - 1} = x^{-1 / 3 - 3 / 3} = x^{-4 / 3} \]. Therefore, \[ \frac{d}{dx} x^{-1 / 3} = -\frac{1}{3} x^{-4 / 3} \].

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Key Concepts

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

Power Rule
The power rule is a fundamental tool in calculus for finding derivatives of functions that are powers of x. It simplifies the differentiation process and is given by the formula: \[ \frac{d}{dx} x^n = nx^{n-1} \]. Here, $$n$$ represents the exponent. Using this rule, if you know the exponent of x, you can easily find its derivative by:
  • Multiplying the function by the exponent $$n$$.
  • Subtracting one from the exponent.
For example: If you have the function \[ f(x) = x^{4} \], its derivative would be: \[ f'(x) = 4x^3 \]. This process works even when the exponent is a fraction or a negative number.
Exponent Rules
Understanding exponent rules is crucial for applying the power rule correctly. Exponent rules help simplify complex expressions and manage different powers of variables. Here are some key rules you should know:
  • **Product Rule for Exponents**: $$ x^a \times x^b = x^{a+b} $$
  • **Quotient Rule for Exponents**: $$ \frac{x^a}{x^b} = x^{a-b} $$
  • **Power of a Power Rule**: $$ (x^a)^b = x^{a \times b} $$
For example, if you have the expression $$ x^2 \times x^3 $$, you would add the exponents to get $$ x^5 $$.
Simplification of Expressions
Simplification is an essential step in calculus to make expressions easier to work with. It involves combining like terms and reducing complex exponents. Let's see how to simplify the derivative we calculated earlier: The expression \[ \frac{d}{dx} \left( x^{-1/3} \right) = -\frac{1}{3} x^{-4/3} \] involves simplifying the exponent. To do this:
  • Combine the exponents: $$ -1/3 - 1 $$
  • Convert 1 to a fraction: $$ -1/3 - 3/3 = -4/3 $$
This gives us a simplified exponent of \[ x^{-4/3} \]. Always ensure your final expression is in its simplest form for clarity and accuracy.

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Most popular questions from this chapter

Use limits to compute the following derivatives. $$f^{\prime}(0), \text { where } f(x)=x^{3}+3 x+1$$

Function The revenue from producing (and selling) \(x\) units of a product is given by \(R(x)=3 x-.01 x^{2}\) dollars. (a) Find the marginal revenue at a production level of \(20 .\) (b) Find the production levels where the revenue is 200 dollars.

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(1), \text { where } f(x)=\frac{1}{1+x^{2}}$$

Apply the three-step method to compute the derivative of the given function. $$f(x)=3 x^{2}-2$$

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{5 x+3}{3 x-2}$$
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