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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 f(x)=c where c is a constant, then f(x)=0

Proof: f(x)=limh0f(x+h)f(x)h=limh0cch=0

Theorem: If c is a constant and f is differentiable at all x, then ddx[cf(x)]=cddx[f(x)]. In simpler notation (cf)=c(f)=cf

The Power Rule

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

Example A

Find f(x) for f(x)=16


If f(x)=16 for all x, then f(x)=0 for all x
We can also write ddx16=0

Example B

Find the derivative of f(x)=4x3


ddx[4x3] ..... Restate the function
4ddx[x3] ..... Apply the Commutative Law
4[3x2] ..... Apply the Power Rule
12x2 ..... Simplify

Example C

Find the derivative of f(x)=2x4


ddx[2x4] ..... Restate
ddx[2x4] ..... Rules of exponents
2ddx[x4] ..... By the Commutative law
2[4x41] ..... Apply the Power Rule
2[4x5] ..... Simplify
8x5 ..... Simplify again
8x5 ..... 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) f(x)=x3

2) f(x)=x

3) f(x)=x

4) f(x)=1x3


1) By the power rule:

If f(x)=x3 then f(x)=(3)x32=3x2

2) Special application of the power rule:


3) Restate the function: ddx[x]

Using rules of exponents (from Algebra): ddx[x1/2]
Apply the Power Rule: 12x1/21
Simplify: 12x1/2
Rules of exponents: 12x1/2
Simplify: 12x

4) Restate the function: ddx[1x3]

Rules of exponents: ddx[x3]
Power Rule: 3x31
Simplify: 3x4
Rules of exponents: 3x4


  1. State the Power Rule.

Find the derivative:

  1. y=5x7
  2. y=3x
  3. f(x)=13x+43
  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:
At Grade
Date Created:
Nov 01, 2012
Last Modified:
May 26, 2016
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