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# 8.9: Constant Derivatives and the Power Rule

Difficulty Level: At Grade Created by: CK-12

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!

### 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 $f(x) = c$ where c is a constant, then $f^{\prime}(x) = 0$ .

Proof: $f'(x)= \lim_{h \to 0}\frac {f(x+h)-f(x)}{h}=\lim_{h \to 0}\frac{c-c}{h} = 0$ .

Theorem: If $c$ is a constant and $f$ is differentiable at all $x$ , then $\frac {d}{dx}[cf(x)] = c\frac {d}{dx}[f(x)]$ . 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
$\frac {d}{dx}[x^n] = nx^{n-1}$ .

#### Example A

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 $\frac{d}{dx}16 = 0$ .

#### Example B

Find the derivative of $f(x)=4x^3$ .

Solution:

$\frac {d}{dx}\left [{4x^3} \right]$ ..... Restate the function
$4 \frac{d}{dx}\left [{x^3} \right]$ ..... Apply the commutative law
$4 \left [{3x^2} \right]$ ..... Apply the power Rule
$12x^2$ ..... Simplify

#### Example C

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

Solution:

$\frac {d}{dx} \left [\frac{-2}{x^4} \right]$ ..... Restate
$\frac {d}{dx}\left [{-2x^{-4}} \right]$ ..... Rules of exponents
$-2 \frac {d}{dx}\left [{x^{-4}} \right]$ ..... By the commutative law
$-2 \left [{-4x^{-4-1}} \right]$ ..... Apply the power rule
$-2 \left [{-4x^{-5}} \right]$ ..... Simplify
$8x^{-5}$ ..... Simplify again
$\frac {8}{x^5}$ ..... Use rules of exponents

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### Guided Practice

Find the derivatives of:

1) $f(x)=x^{3}$

2) $f(x)=x$

3) $f(x)=\sqrt{x}$

4) $f(x)=\frac{1}{x^{3}}$

1) By the power rule:

If $f(x) = x^3$ then $f(x) = (3)x^{3-1} = 3x^2$ .

2) Special application of the power rule:

$\frac {d}{dx}[x] = 1x^{1-1} = x^0 = 1$

3) Restate the function: $\frac {d}{dx}[\sqrt{x}]$

Using rules of exponents (from algebra): $\frac {d}{dx}[x^{1/2}]$
Apply the power rule: $\frac {1}{2}x^{1/2-1}$
Simplify: $\frac {1}{2}x^{-1/2}$
Rules of exponents: $\frac{1}{2x^{1/2}}$
Simplify: $\frac {1}{2\sqrt{x}}$

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

Rules of exponents: $\frac {d}{dx}\left [{x^{-3}} \right ]$
Power rule: $-3x^{-3-1}$
Simplify: $-3x^{-4}$
Rules of exponents: $\frac {-3}{x^4}$

### Explore More

1. State the power rule.

Find the derivative:

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

Nov 01, 2012

Feb 26, 2015

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