Section 2.3
Table of Contents
Differentiation Formulas
Differentiation Formulas #
For each of these differentiation formulas, we’ll give it using Leibniz notation. We’ll often use the Newton/prime notation, and if at any time you have a question, please ask!
Formulas #
 Constant function: $\dfrac{d}{dx} (c) = 0$ if $c$ is a constant. The slope of the line $y=c$ is 0.
 Basic linear function: $\dfrac{d}{dx} (x) = 1$ The slope of the line $y=x$ is 1.
 The Power Rule $\dfrac{d}{dx} (x^n) = nx^{n1}$ if $n$ is a nonzero integer.
Examples #
 Find the derivative of $y = x^3$
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$\dfrac{dy}{dx} = 3 x^{2}  Find the derivative of $y = x^{1551}$
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$\dfrac{dy}{dx} = 1551 x^{1550}$  Find the derivative of $f(x) = \sqrt{x}$
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$f'(x) = \dfrac{1}{2} x^{1/2} = \dfrac{1}{2x^{1/2}} = \dfrac{1}{2\sqrt{x}} $  Find the derivative of $f(x) = \frac{1}{x^5} = x^{5}$
Click to reveal the answer.
$f'(x)= 5 x^{51} = \dfrac{5}{x^{6}}$
Formulas, continued #

Constant multiple rule: if $c$ is a constant, $$ \dfrac{d}{dx} (c f(x) ) = c \dfrac{d}{dx} f(x) $$

Sum Rule $$ \dfrac{d}{dx} (f(x) + g(x) ) = \dfrac{d}{dx} f(x) + \dfrac{d}{dx} g(x) $$

Difference Rule $$ \dfrac{d}{dx} (f(x)  g(x) ) = \dfrac{d}{dx} f(x)  \dfrac{d}{dx} g(x) $$
Examples #
 Find the derivative of $f(x) = x^3  x + 1$
Click to reveal the answer.
$f'(x) = 3x^2  1 + 0 = 3x^2  1$, just like we found in a previous section!
Formulas, continued #

$\dfrac{d}{dx} (\sin x) = \cos x$

$\dfrac{d}{dx} (\cos x) =  \sin x$
Examples: #

Find the derivative of $g(x) = 2\sin x  3\cos x$
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$g'(x) = 2\cos x  3\cdot (\sin x) = 2\cos x + 3 \sin x$ 
If $f(x) = \sin x$, find $f’(x)$, $f’’(x)$, $f’’’(x)$, and $f^{(4)}(x)$
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$\begin{align*} f(x) &= \sin x \\ f'(x) &= \cos x \\ f''(x) &=  \sin x \\ f^{(4)}(x) &= \cos x \\ f^{(5)}(x) &=  (\sin x) = \sin x & \text{ not that I asked...}\end{align*}$ What pattern do you notice? What do you think $f^{(6)}(x)$ is? What about $f^{(25)}(x)$?
Formulas, continued, #
I’m going to switch to Newtonian/prime notation for these because it’s much cleaner and clearer to write and read.

The Product Rule. $$ (fg)’ = f’g + fg' $$
EXAMPLE Find the derivative of $f(x) = x\sin x$
 The Quotient Rule. $$ \left[ \dfrac{f}{g} \right]’ = \dfrac{gf’  fg’}{g^2} $$ EXAMPLE Find the derivative of $h(x) = \dfrac{x^2}{\cos x}$
Examples #
 Compute the derivative of $f(x) = \tan x$
 Compute the derivative of $f(x) = \dfrac{1}{x^5}$ using quotient rule.
 Compute the derivative of $f(x) = \dfrac{1}{x^5}$ using the product rule.
Now practice! #
Head over to WebAssign and work on section 2.3. If you have questions, let me know!