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8.9: Constant Derivatives and the Power Rule

Difficulty Level: At Grade Created by: CK-12
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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!

Watch This

Embedded Video:

- Khan Academy: Calculus: Derivatives 3

Guidance

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 \begin{align*}f(x) = c\end{align*}f(x)=c where c is a constant, then \begin{align*}f^{\prime}(x) = 0\end{align*}

Proof: \begin{align*}f'(x)= \lim_{h \to 0}\frac {f(x+h)-f(x)}{h}=\lim_{h \to 0}\frac{c-c}{h} = 0\end{align*}

Theorem: If \begin{align*}c\end{align*} is a constant and \begin{align*}f\end{align*} is differentiable at all \begin{align*}x\end{align*}, then \begin{align*}\frac {d}{dx}[cf(x)] = c\frac {d}{dx}[f(x)]\end{align*}. In simpler notation \begin{align*}(cf)^{\prime} = c(f)^{\prime} = cf^{\prime}\end{align*}

The Power Rule

Theorem: (The Power Rule) If n is a positive integer, then for all real values of x
\begin{align*}\frac {d}{dx}[x^n] = nx^{n-1}\end{align*}.

Example A

Find \begin{align*}f^{\prime} (x)\end{align*} for \begin{align*}f(x)=16\end{align*}

Solution

If \begin{align*}f(x) = 16\end{align*} for all \begin{align*}x\end{align*}, then \begin{align*}f^{\prime} (x) = 0\end{align*} for all \begin{align*}x\end{align*}
We can also write \begin{align*}\frac{d}{dx}16 = 0\end{align*}

Example B

Find the derivative of \begin{align*}f(x)=4x^3\end{align*}

Solution

\begin{align*}\frac {d}{dx}\left [{4x^3} \right]\end{align*} ..... Restate the function
\begin{align*}4 \frac{d}{dx}\left [{x^3} \right]\end{align*} ..... Apply the Commutative Law
\begin{align*}4 \left [{3x^2} \right]\end{align*} ..... Apply the Power Rule
\begin{align*}12x^2\end{align*} ..... Simplify

Example C

Find the derivative of \begin{align*}f(x)=\frac{-2}{x^{4}}\end{align*}

Solution

\begin{align*}\frac {d}{dx} \left [\frac{-2}{x^4} \right]\end{align*} ..... Restate
\begin{align*}\frac {d}{dx}\left [{-2x^{-4}} \right]\end{align*} ..... Rules of exponents
\begin{align*}-2 \frac {d}{dx}\left [{x^{-4}} \right]\end{align*} ..... By the Commutative law
\begin{align*}-2 \left [{-4x^{-4-1}} \right]\end{align*} ..... Apply the Power Rule
\begin{align*}-2 \left [{-4x^{-5}} \right]\end{align*} ..... Simplify
\begin{align*}8x^{-5}\end{align*} ..... Simplify again
\begin{align*}\frac {8}{x^5}\end{align*} ..... Use rules of exponents

Vocabulary

A theorem is a statement accepted to be true based on a series of reasoned statements already accepted to be true. In the context of this lesson, a theorem is a rule that allows a quick calculation of the derivative of functions of different types.

A proof is a series of true statements leading to the acceptance of truth of a more complex statement.

Guided Practice

Questions

Find the derivatives of:

1) \begin{align*}f(x)=x^{3}\end{align*}

2) \begin{align*}f(x)=x\end{align*}

3) \begin{align*}f(x)=\sqrt{x}\end{align*}

4) \begin{align*}f(x)=\frac{1}{x^{3}}\end{align*}

Solutions

1) By the power rule:

If \begin{align*}f(x) = x^3\end{align*} then \begin{align*}f(x) = (3)x^{3-2} = 3x^2\end{align*}

2) Special application of the power rule:

\begin{align*}\frac {d}{dx}[x] = 1x^{1-1} = x^0 = 1\end{align*}

3) Restate the function: \begin{align*}\frac {d}{dx}[\sqrt{x}]\end{align*}

Using rules of exponents (from Algebra): \begin{align*}\frac {d}{dx}[x^{1/2}]\end{align*}
Apply the Power Rule: \begin{align*}\frac {1}{2}x^{1/2-1}\end{align*}
Simplify: \begin{align*}\frac {1}{2}x^{-1/2}\end{align*}
Rules of exponents: \begin{align*}\frac{1}{2x^{1/2}}\end{align*}
Simplify: \begin{align*}\frac {1}{2\sqrt{x}}\end{align*}

4) Restate the function: \begin{align*}\frac {d}{dx}\left [ \frac{1}{x^3} \right ]\end{align*}

Rules of exponents: \begin{align*}\frac {d}{dx}\left [{x^{-3}} \right ]\end{align*}
Power Rule: \begin{align*}-3x^{-3-1}\end{align*}
Simplify: \begin{align*}-3x^{-4}\end{align*}
Rules of exponents: \begin{align*}\frac {-3}{x^4}\end{align*}

Practice

  1. State the Power Rule.

Find the derivative:

  1. \begin{align*}y = 5x^7\end{align*}
  2. \begin{align*}y = -3x\end{align*}
  3. \begin{align*}f(x) = \frac{1} {3} x + \frac{4} {3}\end{align*}
  4. \begin{align*}y = x^4 - 2x^3 - 5\sqrt{x} + 10\end{align*}
  5. \begin{align*}y = (5x^2 - 3)^2\end{align*}
  6. given \begin{align*}y(x)= x^{-4\pi^2}\end{align*} when \begin{align*} x = 1\end{align*}
  7. y(x) = 5</math>
  8. given \begin{align*}u(x)= x^{-5\pi^3}\end{align*} what is \begin{align*} u'(2)\end{align*}
  9. \begin{align*} y = \frac{1}{5}\end{align*} when \begin{align*} x = 4 \end{align*}
  10. given \begin{align*}d(x)= x^{-0.37}\end{align*} what is \begin{align*} d'(1)\end{align*}
  11. \begin{align*} g(x) = x^{-3}\end{align*}
  12. \begin{align*}u(x) = x^{0.096}\end{align*}
  13. \begin{align*}k(x) = x{-0.49}\end{align*}
  14. \begin{align*} y = x^{-5\pi^3}\end{align*}

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Vocabulary

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'(x), \frac{dy}{dx}, y', \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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