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Chapter 1: Problem 44

Find \(f^{\prime}(x)\). $$f(x)=\frac{x^{3 / 2}}{3}$$

Short Answer

Expert verified
\( f^{\prime}(x) = \frac{1}{2} x^{1/2} \)

Step by step solution

01

- Recognize the form of the function

The function given is a fraction involving a power of x: \[ f(x)=\frac{x^{3 / 2}}{3} \]
02

- Simplify the function

Rewrite the function in a simpler form by factoring the constant outside of the derivative: \[ f(x) = \frac{1}{3} x^{3/2} \]
03

- Differentiate the function

Use the power rule for differentiation, where \( \frac{d}{dx} x^n = nx^{n-1} \). Apply this to the simplified function: \[ f^{\prime}(x) = \frac{1}{3} \cdot \frac{d}{dx}(x^{3/2}) \] The derivative of \( x^{3/2} \) is \( \frac{3}{2} x^{(3/2)-1} \): \[ \frac{d}{dx}(x^{3/2}) = \frac{3}{2} x^{1/2} \]
04

- Combine the results

Combine the constant factor with the derivative of \( x^{3/2} \): \[ f^{\prime}(x) = \frac{1}{3} \cdot \frac{3}{2} x^{1/2} \] Simplify the result: \[ f^{\prime}(x) = \frac{1}{2} x^{1/2} \]

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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 key concept in differentiation, especially for functions that are powers of x. The rule states:
  • For any real number n, if you have a function of the form



    • f(x) = x^n, the derivative is given by:
          \[ \frac{d}{dx} x^n = nx^{n-1} \]
          If your function involves a constant multiplied by a power of x, just differentiate the power part and multiply the constant later. Let's look at our example:
            For \[ f(x) = \frac{1}{3} x^{3/2}, \]
              the power part is .
              x^3/2

            According to the power rule, the derivative of .
            x^{3/2}
              will be:

              \[\frac{3}{2} x^{(3/2)-1} = \frac{3}{2} x^{1/2}. \]

            This result will be multiplied by the constant
            1/3.
              \[ f^{\prime}(x) = \frac{1}{3} \times \frac{3}{2} x^{1/2}\].

            Don鈥檛 forget the key points:
              \br you bring the exponent down as a
              factor
              you
              subtract
              1 from the exponent for the new power.
Simplifying Functions
Simplifying functions can greatly help when you need to differentiate or integrate them. It makes the process more straightforward and error-free. In the given example, simplifying is easy because of the constant fraction:
    The
    original
    function
    given
    was:
      \[\frac{x^{3 / 2}}{3}. \]

    wesimplifiedthis
    to:

      the
      property
      of
      constants
      in
      differentiation,

      meaning you
      can factor
      out
      the
      constant:

        \[ f(x) = \frac{1}{3} x^{3/2}. \]

        This
        simplified
        form
        is
        easier
        to work
        with
        because
        now
        you
        are only
        focusing
        on
        differentiating
        \br> \[ x^3/2
        .
          And
          not
          the
          constant\].
Differentiation
Differentiation
is the
    of finding the derivative of a function, which tells us how the function value changes as .the changing processes
    vary.
      the
      simplest is to apply power
      rule.


            in
            our
            example,
            it
            went
            as

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Most popular questions from this chapter

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2}$$

Consumer demand. Suppose that the demand function for a product is given by \(D(p)=\frac{80,000}{p}\) and that price \(p\) is a function of time given by \(p=1.6 t+9,\) where \(t\) is in days. a) Find the demand as a function of time \(t\) b) Find the rate of change of the quantity demanded when \(t=100\) days.

Use the derivative to help show whether each function is always increasing, always decreasing, or neither. $$f(x)=\sqrt{x}, \quad x \geq 0$$

Tongue-Tied Sauces, Inc., finds that the revenue, in dollars, from the sale of \(x\) bottles of barbecue sauce is given by \(R(x)=7.5 x^{0.7} .\) Find the rate at which average revenue is changing when 81 bottles of barbecue sauce have been produced.

Differentiate. $$y=\left(\frac{x}{\sqrt{x-1}}\right)^{3}$$
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