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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!

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- Khan Academy: Calculus: Derivatives 3


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*}f(x)=0

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*}f(x)=limh0f(x+h)f(x)h=limh0cch=0

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

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*}ddx[xn]=nxn1.

Example A

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


If \begin{align*}f(x) = 16\end{align*}f(x)=16 for all \begin{align*}x\end{align*}x, then \begin{align*}f^{\prime} (x) = 0\end{align*}f(x)=0 for all \begin{align*}x\end{align*}x
We can also write \begin{align*}\frac{d}{dx}16 = 0\end{align*}ddx16=0

Example B

Find the derivative of \begin{align*}f(x)=4x^3\end{align*}f(x)=4x3


\begin{align*}\frac {d}{dx}\left [{4x^3} \right]\end{align*}ddx[4x3] ..... Restate the function
\begin{align*}4 \frac{d}{dx}\left [{x^3} \right]\end{align*}4ddx[x3] ..... Apply the Commutative Law
\begin{align*}4 \left [{3x^2} \right]\end{align*}4[3x2] ..... Apply the Power Rule
\begin{align*}12x^2\end{align*}12x2 ..... Simplify

Example C

Find the derivative of \begin{align*}f(x)=\frac{-2}{x^{4}}\end{align*}f(x)=2x4


\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


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


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*}


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*}


  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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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}.


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


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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Difficulty Level:
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Date Created:
Nov 01, 2012
Last Modified:
May 26, 2016
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