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8.3.1: Constant Derivatives and the Power Rule

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    14820
  • Constant Derivatives and the Power Rule

    The power rule is a fantastic "shortcut" for finding the derivatives of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort - often in your head!


    Constant Derivatives and the Power Rule

    In this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.

    The Derivative of a Constant

    Theorem: If f(x)=c where c is a constant, then f′(x)=0.

    Proof: \(\ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{c-c}{h}=0\).

    Theorem: If \(\ c\) is a constant and \(\ f\) is differentiable at all \(\ x\), then \(\ \frac{d}{d x}[c f(x)]=c \frac{d}{d x}[f(x)]\). 

    In simpler notation \(\ (cf)^{\prime}=c(f)^{\prime}=cf^{\prime}\)

    The Power Rule

    Theorem: (The Power Rule) If n is a positive integer, then for all real values of x

    \(\ \frac{d}{d x}\left[x^{n}\right]=n x^{n-1}\).


    Examples

    Example 1

    Find \(\ f^{\prime}(x)\) for \(\ f(x)=16\).

    Solution

    If \(\ f(x)=16\) for all x, then \(\ f^{\prime}(x)=0\) for all x.

    We can also write \(\ \frac{d}{d x} 16=0\).

    Example 2

    Find the derivative of \(\ f(x)=4 x^{3}\).

    Solution

    \(\ \frac{d}{d x}\left[4 x^{3}\right]\) ..... Restate the function

    \(\ 4 \frac{d}{d x}\left[x^{3}\right]\) ..... Apply the commutative law

    \(\ 4\left[3 x^{2}\right]\) ..... Apply the power Rule

    \(\ 12 x^{2}\) ..... Simplify

    Example 3

    Find the derivative of \(\ f(x)=\frac{-2}{x^{4}}\).

    Solution

    \(\ \frac{d}{d x}\left[\frac{-2}{x^{4}}\right]\) ..... Restate

    \(\ \frac{d}{d x}\left[-2 x^{-4}\right]\) ..... Rules of exponents

    \(\ -2 \frac{d}{d x}\left[x^{-4}\right]\) ..... By the commutative law

    \(\ -2\left[-4 x^{-4-1}\right]\) ..... Apply the power rule

    \(\ -2\left[-4 x^{-5}\right]\) ..... Simplify

    \(\ 8 x^{-5}\) ..... Simplify again

    \(\ \frac{8}{x^{5}}\) ..... Use rules of exponents

    Example 4

    Find the derivative of \(\ f(x)=x\).

    Solution

    Special application of the power rule:

    \(\ \frac{d}{d x}[x]=1 x^{1-1}=x^{0}=1\)

    Example 5

    Find the derivative of \(\ f(x)=\sqrt{x}\).

    Solution

    Restate the function: \(\ \frac{d}{d x}[\sqrt{x}]\)

    Using rules of exponents (from algebra): \(\ \frac{d}{d x}\left[x^{1 / 2}\right]\)

    Apply the power rule: \(\ \frac{1}{2} x^{1 / 2-1}\)

    Simplify: \(\ \frac{1}{2} x^{-1 / 2}\)

    Rules of exponents: \(\ \frac{1}{2 x^{1 / 2}}\)

    Simplify: \(\ \frac{1}{2 \sqrt{x}}\)

    Example 6

    Find the derivative of \(\ f(x)=\frac{1}{x^{3}}\).

    Solution

    Restate the function: \(\ \frac{d}{d x}\left[\frac{1}{x^{3}}\right]\)

    Rules of exponents: \(\ \frac{d}{d x}\left[x^{-3}\right]\)

    Power rule: \(\ -3 x^{-3-1}\)

    Simplify: \(\ -3 x^{-4}\)

    Rules of exponents: \(\ \frac{-3}{x^{4}}\)


    Review

    1. State the power rule.

    Find the derivative:

    1. \(\ y=5 x^{7}\)
    2. \(\ y=-3 x\)
    3. \(\ f(x)=\frac{1}{3} x+\frac{4}{3}\)
    4. \(\ y=x^{4}-2 x^{3}-5 \sqrt{x}+10\)
    5. \(\ y=\left(5 x^{2}-3\right)^{2}\)
    6. Given \(\ y(x)=x^{-4 \pi^{2}}\), find the derivative when \(\ x=1\).
    7. \(\ y(x)=5\)
    8. Given \(\ u(x)=x^{-5 \pi^{3}}\), what is \(\ u^{\prime}(2)\)?
    9. \(\ y=\frac{1}{5}\) when \(\ x=4\)
    10. Given \(\ d(x)=x^{-0.37}\), what is \(\ d^{\prime}(1)\)?
    11. \(\ g(x)=x^{-3}\)
    12. \(\ u(x)=x^{0.096}\)
    13. \(\ k(x)=x-0.49\)
    14. \(\ y=x^{-5 \pi^{3}}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 8.9.


    Vocabulary

    Term Definition
    derivative The derivative of a function is the slope of the line tangent to the function at a given point on the graph. Notations for derivative include \(\ f^{\prime}(x), \frac{d y}{dx}, y^{\prime}, \frac{df}{dx}\) and \(\ \frac{df(x)}{dx}\).
    proof A proof is a series of true statements leading to the acceptance of truth of a more complex statement.
    theorem A theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.
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