The purpose of this lesson is to begin to develop shortcuts for finding derivatives.

Here's some basic information about derivatives:

• 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.
• The derivative of $\,f\,$ is named $\,f'\,$ (using prime notation).
  The function $\,f'\,$ is read aloud as ‘$\,f\,$ prime’.

• 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.
• 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 $\,\bigl(x,f(x)\bigr)\,$’.
• 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 $\,\bigl(x,f(x)\bigr)\,.$
• 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
  • 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
• Every important calculus idea is defined in terms of a limit.
  For example, the derivative is defined in terms of a limit.
• By definition, when the limit exists, $\,\displaystyle f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}\,.$
• 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.
• 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

• 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 $\,g'(t) = 0$
  • $\displaystyle\frac{d}{dx}(\pi) = 0\,,$     $\displaystyle\frac{d}{dt}(\sqrt{7}) = 0\,,$     $\displaystyle\frac{d}{dw}({\text{e}}^3) = 0\,$
• 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:
  • if $g(t) = 5t - 1\,,$ then $\,g'(t) = 5\,$
  • $\displaystyle\frac{d}{dx} (3 - \frac{2}{7}x) = -\frac 27\,$;
    
    note that $\,y = 3 - \frac 27x = -\frac 27x + 3\,$ graphs as a line with slope $\,-\frac 27\,.$

• Power functions are functions that can be written in the form $\,x^n\,,$ for some real number $\,n\,.$
  Examples of power functions:
  • $x^2\,,$   $x^3\,,$   $x^{1.4}\,,$   $x^\pi$
  • $\displaystyle\frac 1x = x^{-1}\,,$   $\displaystyle\frac 1{x^2} = x^{-2}\,,$   $\sqrt x = x^{1/2}\,,$   $\root 3\of {x^2} = x^{2/3}$
  The Simple Power Rule tells how to differentiate power functions.
• To differentiate $\,x^n\,$:
  • bring the exponent down front;
  • decrease the original exponent by $\,1$
• EXAMPLES:
  
  • if $f(x) = x^3\,,$ then $f'(x) = 3x^2$
  • $\displaystyle \frac{d}{dx} \sqrt x = \frac d{dx} x^{1/2} = \frac{1}{2} x^{-1/2} = \frac{1}{2x^{1/2}} = \frac{1}{2\sqrt x}$
  • $\displaystyle \frac{d}{dt} \frac{1}{\sqrt t} = \frac d{dt} t^{-1/2} = -\frac{1}{2} t^{-3/2} = -\frac{1}{2t^{3/2}} = \frac{-1}{2\sqrt {t^3}}$
  • $\displaystyle \frac{d}{dt} t\root 3\of t = \frac{d}{dt} t^1 t^{1/3} = \frac d{dt} t^{4/3} = \frac 43 t^{1/3} = \frac 43\root 3\of t = \frac {4\root 3\of t}{3}$