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# Constant Derivatives and the Power Rule

## Derivative of a constant is zero and \frac {d}{dx}[x^n] = nx^{n-1} .

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

Proof: f(x)=limh0f(x+h)f(x)h=limh0cch=0\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 c\begin{align*}c\end{align*} is a constant and f\begin{align*}f\end{align*} is differentiable at all x\begin{align*}x\end{align*}, then ddx[cf(x)]=cddx[f(x)]\begin{align*}\frac {d}{dx}[cf(x)] = c\frac {d}{dx}[f(x)]\end{align*}. In simpler notation (cf)=c(f)=cf\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
ddx[xn]=nxn1\begin{align*}\frac {d}{dx}[x^n] = nx^{n-1}\end{align*}.

### Examples

#### Example 1

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

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

We can also write ddx16=0\begin{align*}\frac{d}{dx}16 = 0\end{align*}.

#### Example 2

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

ddx[4x3]\begin{align*}\frac {d}{dx}\left [{4x^3} \right]\end{align*} ..... Restate the function

4ddx[x3]\begin{align*}4 \frac{d}{dx}\left [{x^3} \right]\end{align*} ..... Apply the commutative law

4[3x2]\begin{align*}4 \left [{3x^2} \right]\end{align*} ..... Apply the power Rule

12x2\begin{align*}12x^2\end{align*} ..... Simplify

#### Example 3

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

ddx[2x4]\begin{align*}\frac {d}{dx} \left [\frac{-2}{x^4} \right]\end{align*} ..... Restate

ddx[2x4]\begin{align*}\frac {d}{dx}\left [{-2x^{-4}} \right]\end{align*} ..... Rules of exponents

2ddx[x4]\begin{align*}-2 \frac {d}{dx}\left [{x^{-4}} \right]\end{align*} ..... By the commutative law

2[4x41]\begin{align*}-2 \left [{-4x^{-4-1}} \right]\end{align*} ..... Apply the power rule

2[4x5]\begin{align*}-2 \left [{-4x^{-5}} \right]\end{align*} ..... Simplify

8x5\begin{align*}8x^{-5}\end{align*} ..... Simplify again

8x5\begin{align*}\frac {8}{x^5}\end{align*} ..... Use rules of exponents

#### Example 4

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

Special application of the power rule:

ddx[x]=1x11=x0=1\begin{align*}\frac {d}{dx}[x] = 1x^{1-1} = x^0 = 1\end{align*}

#### Example 5

Find the derivative of f(x)=x\begin{align*}f(x)=\sqrt{x}\end{align*}.

Restate the function: ddx[x]\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*}

#### Example 6

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

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*}

### Review

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*}, find the derivative when \begin{align*} x = 1\end{align*}.
7. \begin{align*}y(x) = 5\end{align*}
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 Language: English

TermDefinition
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.