Title

$\underset{⏟}{\begin{matrix} If f (x) = a constant, then f^{'} (x) = 0 \end{matrix}} \begin{array}{r} a constant is something without x \\ f^{'} (x) measures how fast f (x) changes as x changes \\ A constant doesn't change as x changes, so it's derivative is 0 \\ f^{'} (x) is the slope of y = f (x) \\ y = a constant is a horizontal line, so it has slope 0 \\ Therefore if f (x) = a constant, f^{'} (x) = 0 \end{array}$

$\underset{⏟}{\begin{matrix} This rule can be abbreviated as (c)^{'} = 0 \end{matrix}} \begin{array}{r} We replaced the constant with c \\ Since f (x) = c, we can replace f^{'} (x) with (c)^{'} \end{array}$

Ex 1: $f (x) = 4$ . Find $f^{'} (x)$


	$Step 1: \underset{⏟}{\begin{matrix} If f (x) = a constant, then f^{'} (x) = 0 \end{matrix}} \begin{array}{r} a constant is something without x \\ f^{'} (x) measures how fast f (x) changes as x changes \\ A constant doesn't change as x changes, so it's derivative is 0 \end{array}$	$f^{'} (x) = 0$

Subsection 5.2.2: Derivative of

c x

$\underset{⏟}{\begin{matrix} If f (x) = c x, then f^{'} (x) = c \end{matrix}} \begin{array}{r} c is something without x \\ If there is no c, c = 1 \\ f^{'} (x) is the slope of f (x) . The slope of y = c x is c \\ Therefore if f (x) = c x, then f^{'} (x) = c \end{array}$

Ex 1: $f (x) = 4 x$ . Find $f^{'} (x)$


	$Step 1: \underset{⏟}{\begin{matrix} If f (x) = c x, then f^{'} (x) = c \end{matrix}} \begin{array}{r} c is something without x \\ If there is no c, c = 1 \end{array}$	$\underset{⏟}{\begin{matrix} f^{'} (x) = 4 \end{matrix}} \begin{array}{r} c = 4 b/c f (x) = 4 x \end{array}$

Subsection 5.2.3: Derivative of

c x^{n}

Ex 1: $f (x) = 4 x^{3}$ . Find $f^{'} (x)$


	$Step 1: \underset{⏟}{\begin{matrix} If f (x) = c x^{n}, then f^{'} (x) = n \cdot c x^{n - 1} \end{matrix}} \begin{array}{r} c is something without x . If there is no c, c = 1 \\ If there is no n, n = 1, but it is better to use the rule (c x)^{'} = c \end{array}$	$\underset{⏟}{\begin{matrix} f^{'} (x) = 12 x^{2} \end{matrix}} \begin{array}{r} c = 4 b/c f (x) = 4 x^{3} \\ n = 3 b/c f (x) = 4 x^{3} \\ \begin{aligned} f^{'} (x) & = 3 \cdot 4 x^{3 - 1} \\ = 12 x^{2} \end{aligned} \end{array}$

Subsection 5.2.4: Derivatives with Properties of Exponents

There are three properties of exponents that are useful when calculating derivatives of fractions and roots

\begin{aligned} \underset{⏟}{\begin{array}{c} \frac{c}{x^{n}} = c x^{- n} \end{array}} \begin{array}{r} Moving the power out of the \\ denominator makes it negative \end{array} & \underset{⏟}{\begin{array}{c} c \sqrt[n]{x} = c x^{\frac{1}{n}} \end{array}} \begin{array}{r} The root becomes a \\ fraction power \end{array} & \underset{⏟}{\begin{array}{c} \frac{c}{\sqrt[n]{x}} = c x^{- \frac{1}{n}} \end{array}} \begin{array}{r} The root becomes a fraction power and \\ moving the root outside of the \\ denominator makes the power negative \\ \frac{c}{\sqrt[n]{x}} = \frac{c}{x^{\frac{1}{n}}} = c x^{- \frac{1}{n}} \end{array} \\ \underset{⏟}{\begin{array}{c} \frac{c}{x} = c x^{- 1} \end{array}} \begin{array}{r} There is a invisible power of 1 \\ Moving the power outside of the \\ denominator makes the exponent negative \\ \frac{c}{x} = \frac{c}{x^{1}} = c x^{- 1} \end{array} & \underset{⏟}{\begin{array}{c} c \sqrt{x} = c x^{\frac{1}{2}} \end{array}} \begin{array}{r} This is the same property as \\ c \sqrt[n]{x} = c x^{\frac{1}{n}} with n = 2 \\ because if there is no n in \\ a square root, n = 2 \end{array} & \underset{⏟}{\begin{array}{c} \frac{c}{\sqrt{x}} = c x^{- \frac{1}{2}} \end{array}} \begin{array}{r} This is the same property as \\ \frac{c}{\sqrt[n]{x}} = c x^{- \frac{1}{n}} with n = 2 \\ because if there is no n in \\ a square root, n = 2 \end{array} \end{aligned}

Ex 1: $f (x) = \frac{2}{\sqrt[3]{x}}$ . Find $f^{'} (x)$


	$Step 1: \underset{⏟}{\begin{matrix} Rewrite f (x) in the form c x^{n} using properties of exponents \end{matrix}} \begin{array}{r} \begin{aligned} \frac{c}{x^{n}} = c x^{- n} & c \sqrt[n]{x} = c x^{\frac{1}{n}} & \frac{c}{\sqrt[n]{x}} = c x^{- \frac{1}{n}} \\ \frac{c}{x} = c x^{- 1} & c \sqrt{x} = c x^{\frac{1}{2}} & \frac{c}{\sqrt{x}} = c x^{- \frac{1}{2}} \end{aligned} \end{array}$	$\underset{⏟}{\begin{matrix} f (x) = 2 x^{- \frac{1}{3}} \end{matrix}} \begin{array}{r} Use \frac{c}{\sqrt[n]{x}} = c x^{- \frac{1}{n}} b/c f (x) = \frac{2}{\sqrt[3]{x}} \\ c = 2 b/c f (x) = \frac{2}{\sqrt[3]{x}} \\ n = 3 b/c f (x) = \frac{2}{\sqrt[3]{x}} \\ \begin{aligned} f (x) & = c x^{- \frac{1}{n}} \\ = 2 x^{- \frac{1}{3}} \end{aligned} \end{array}$
	$Step 2: \underset{⏟}{\begin{matrix} If f (x) = c x^{n}, then f^{'} (x) = n \cdot c x^{n - 1} \end{matrix}} \begin{array}{r} c is something without x . If there is no c, c = 1 \\ If there is no n, n = 1, but it is better to use the rule (c x)^{'} = c \end{array}$	$\underset{⏟}{\begin{matrix} - \frac{1}{3} \cdot 2 x^{- \frac{1}{3} - 1} \end{matrix}} \begin{array}{r} c = 2 b/c f (x) = 2 x^{- \frac{1}{3}} \\ n = - \frac{1}{3} b/c f (x) = 2 x^{- \frac{1}{3}} \\ \begin{aligned} f^{'} (x) & = - \frac{1}{3} \cdot 2 x^{- \frac{1}{3} - 1} \end{aligned} \end{array}$

Subsection 5.2.5: Derivative With Multiple Terms

Ex 1: $f (x) = - 4 \sqrt[3]{x} + x$ . Find $f^{'} (x)$


	$Step 1: \underset{⏟}{\begin{matrix} \begin{matrix} Rewrite fractions and roots \\ using properties of exponents \end{matrix} \end{matrix}} \begin{array}{r} \begin{aligned} \frac{c}{x^{n}} = c x^{- n} & c \sqrt[n]{x} = c x^{\frac{1}{n}} & \frac{c}{\sqrt[n]{x}} = c x^{- \frac{1}{n}} \\ \frac{c}{x} = c x^{- 1} & c \sqrt{x} = c x^{\frac{1}{2}} & \frac{c}{\sqrt{x}} = c x^{- \frac{1}{2}} \end{aligned} \end{array}$	$NA$
	$Step 2: \underset{⏟}{\begin{matrix} \begin{matrix} Calculate the derivative \\ of each term \end{matrix} \end{matrix}} \begin{aligned} \begin{aligned} Use the rules \\ \begin{aligned} (c)^{'} & = 0 \\ (c x)^{'} & = c \\ (c x^{n})^{'} & = n \cdot c x^{n - 1} \end{aligned} \end{aligned} & \begin{array}{c} Don't forget to take the \\ derivative of the terms \\ you rewrote in Step 1 \end{array} \end{aligned}$	$f^{'} (x) = u n d e f i n e d + \underset{⏟}{\begin{matrix} 1 \end{matrix}} \begin{array}{r} Use (c x)^{'} = c \\ c = 1 b/c f (x) = 1 x \\ f^{'} (x) = 1 \end{array}$

Section 5.2: Lecture 9/7Hide/Show in Tooltips

Subsection 5.2.1: Derivative of a Constant

Ex 1: $f (x) = 4$ . Find $f^{'} (x)$

Subsection 5.2.2: Derivative of $c x$

Ex 1: $f (x) = 4 x$ . Find $f^{'} (x)$

Subsection 5.2.3: Derivative of $c x^{n}$

Ex 1: $f (x) = 4 x^{3}$ . Find $f^{'} (x)$

Subsection 5.2.4: Derivatives with Properties of Exponents

Ex 1: $f (x) = \frac{2}{\sqrt[3]{x}}$ . Find $f^{'} (x)$

Subsection 5.2.5: Derivative With Multiple Terms

Ex 1: $f (x) = - 4 \sqrt[3]{x} + x$ . Find $f^{'} (x)$

Section 5.2: Lecture 9/7Hide/Show in Tooltips Disable ColorsHideShow

Subsection 5.2.1: Derivative of a ConstantLinkHideShow

Ex 1: f(x)=4. Find f′(x)HideShow

Subsection 5.2.2: Derivative of cxLinkHideShow

Ex 1: f(x)=4x. Find f′(x)HideShow

Subsection 5.2.3: Derivative of cxnLinkHideShow

Ex 1: f(x)=4x3. Find f′(x)HideShow

Subsection 5.2.4: Derivatives with Properties of ExponentsLinkHideShow

Ex 1: f(x)=2x3. Find f′(x)HideShow

Subsection 5.2.5: Derivative With Multiple TermsLinkHideShow

Ex 1: f(x)=−4x3+x. Find f′(x)HideShow

Section 5.2: Lecture 9/7Hide/Show in Tooltips

Subsection 5.2.1: Derivative of a Constant

Ex 1: $f (x) = 4$ . Find $f^{'} (x)$

Subsection 5.2.2: Derivative of $c x$

Ex 1: $f (x) = 4 x$ . Find $f^{'} (x)$

Subsection 5.2.3: Derivative of $c x^{n}$

Ex 1: $f (x) = 4 x^{3}$ . Find $f^{'} (x)$

Subsection 5.2.4: Derivatives with Properties of Exponents

Ex 1: $f (x) = \frac{2}{\sqrt[3]{x}}$ . Find $f^{'} (x)$

Subsection 5.2.5: Derivative With Multiple Terms

Ex 1: $f (x) = - 4 \sqrt[3]{x} + x$ . Find $f^{'} (x)$