The purpose of this lesson is to begin to develop shortcuts for finding derivatives.
Here's some basic information about derivatives:
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The derivative of a function
$\,f\,$ is a (new) function that gives the
slopes of the
tangent lines
to the graph of $\,f\,.$
Derivatives give slope information.
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The derivative of
$\,f\,$ is named $\,f'\,$ (using prime notation).
The function $\,f'\,$ is read aloud as ‘$\,f\,$ prime’.
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The
$\displaystyle\,\frac{d}{dx}\,$ operator (read as ‘dee dee x’) is an instruction to
‘take the derivative with respect to $\,x\,$ of whatever comes next’.
For example,
$\displaystyle\,\frac{d}{dx}(x^2)\,$ denotes the derivative with respect to $\,x\,$ of $\,x^2\,.$
Similarly, $\displaystyle\,\frac{d}{dt}(t^2)\,$ denotes the derivative with respect to $\,t\,$ of $\,t^2\,.$
The $\displaystyle\,\frac d{dx}\,$ operator is particularly useful when differentiating a function that has not been given a name.
If there is no confusion about the function that $\,\frac{d}{dx}\,$ is acting on, then the parentheses that hold the function
to be differentiated may be dropped.
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If a function $\,f\,$ is differentiable at $\,x\,,$ then there is a non-vertical
tangent line at the point $\,\big(x,f(x)\bigr)\,.$
If there is a non-vertical tangent line at the point $\,\big(x,f(x)\bigr)\,,$
then the function $\,f\,$ is differentiable at $\,x\,.$
Thus, ‘$\,f\,$ is differentiable at $\,x\,$’ is equivalent to
‘there is a non-vertical tangent line to the graph of $\,f\,$ at the point $\,\bigl(x,f(x)\bigr)\,$’.
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The function $f'$, evaluated at $\,x\,,$ is denoted by $\,f'(x)\,.$
‘$\,f'(x)\,$’ is read aloud as ‘$\,f\,$ prime of $\,x\,$’.
The number $\,f'(x)\,$ gives the slope of the tangent line to the graph of $\,f\,$ at the point $\,\bigl(x,f(x)\bigr)\,.$
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Elaborating: for $\,f\,$ to be differentiable at $\,x\,,$ all the following requirements must be met:
- $x$ must be in the domain of $f$, so the point $(x,f(x))$ exists
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there must be a tangent line to the curve at $(x,f(x))$;
the slope of this tangent line captures the ‘direction you're moving’ as you walk along the curve, going from
left to right
- the tangent line must be non-vertical, since a vertical line has no slope
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Every important calculus idea is defined in terms of a limit.
For example, the derivative is defined in terms of a limit.
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By definition, when the limit exists,
$\,\displaystyle f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}\,.$
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The expression $\displaystyle\,\frac{f(x+h) - f(x)}{h}\,$ gives the slope of the line between $\,\bigl(x,f(x)\bigr)\,$ and a nearby
point $\,\bigl(x+h,f(x+h)\bigr)\,.$
By taking the limit as $\,h\,$ approaches $\,0\,,$ the nearby point is ‘slid’ closer and closer to the original point.
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The definition of derivative is tedious to use, particularly as the function $\,f\,$ increases in complexity.
There must be a better way to find $\,f'(x)\,$!
Basic Derivative Shortcuts
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THE DERIVATIVE OF A CONSTANT IS ZERO
A constant function graphs as a horizontal line; the slope of a horizontal line is zero.
Examples:
- if $\,f(x) = 5\,,$ then $\,f'(x) = 0\,$
- if $\,g(t) = \ln 3\,,$ then $\,g'(t) = 0$
- $\displaystyle\frac{d}{dx}(\pi) = 0\,,$ $\displaystyle\frac{d}{dt}(\sqrt{7}) = 0\,,$ $\displaystyle\frac{d}{dw}({\text{e}}^3) = 0\,$
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THE DERIVATIVE OF A LINEAR FUNCTION IS THE SLOPE OF THE LINE
Recall that the graph of $\,f(x) = mx + b\,$ is a line with slope $\,m\,.$
Thus, $\,f'(x) = m\,.$
Examples:
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if $g(t) = 5t - 1\,,$ then $\,g'(t) = 5\,$
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$\displaystyle\frac{d}{dx} (3 - \frac{2}{7}x) = -\frac 27\,$;
note that $\,y = 3 - \frac 27x = -\frac 27x + 3\,$ graphs as a line with slope $\,-\frac 27\,.$
SIMPLE POWER RULE
how to differentiate $\,x^n$
For all real numbers $\,n\,$:
$$\frac d{dx} x^n = nx^{n-1}$$
NOTES ON THE SIMPLE POWER RULE:
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Power functions are functions that can be written in the form $\,x^n\,,$ for some real number $\,n\,.$
Examples of power functions:
- $x^2\,,$ $x^3\,,$ $x^{1.4}\,,$ $x^\pi$
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$\displaystyle\frac 1x = x^{-1}\,,$ $\displaystyle\frac 1{x^2} = x^{-2}\,,$ $\sqrt x = x^{1/2}\,,$
$\root 3\of {x^2} = x^{2/3}$
The Simple Power Rule tells how to differentiate power functions.
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To differentiate $\,x^n\,$:
- bring the exponent down front;
- decrease the original exponent by $\,1$
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EXAMPLES:
- if $f(x) = x^3\,,$ then $f'(x) = 3x^2$
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$\displaystyle \frac{d}{dx} \sqrt x = \frac d{dx} x^{1/2} =
\frac{1}{2} x^{-1/2} = \frac{1}{2x^{1/2}} = \frac{1}{2\sqrt x}$
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$\displaystyle \frac{d}{dt} \frac{1}{\sqrt t} = \frac d{dt} t^{-1/2} =
-\frac{1}{2} t^{-3/2} = -\frac{1}{2t^{3/2}} = \frac{-1}{2\sqrt {t^3}}$
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$\displaystyle \frac{d}{dt} t\root 3\of t = \frac{d}{dt} t^1 t^{1/3} = \frac d{dt} t^{4/3}
= \frac 43 t^{1/3} = \frac 43\root 3\of t = \frac {4\root 3\of t}{3}$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Differentiation Formula Practice