What the h?

Definition

Given a function `f(x)`  we define the derivative of `f(x)`  at `x=a`  as

`f'(a) = lim_(x to a) (f(x)-f(a))/(x-a)` 

Definition

Given a function `f(x)`  we define the derivative of `f(x)`  as

`f'(x) = lim_(h to 0) (f(x+h)-f(x))/h` 

EG

`f(x) = x^2`  find `f'(3)` 

Method 1

`f'(3) = lim_(x to 3) (f(x)-f(3))/(x-3)` 

`=lim_(x to 3) (x^2-3^2)/(x-3)` 

`=lim_(x to 3) ((x+3)(x-3))/(x-3)` 

`=lim_(x to 3) x+3` 

`= 6` 

Method 2

`f'(3) = lim_(h to 0) (f(3+h)-f(3))/h` 

` =lim_(h to 0) ((3+h)^2-3^2)/h` 

` =lim_(h to 0) (9+6h+h^2-9)/h` 

` =lim_(h to 0) (6h+h^2)/h` 

` =lim_(h to 0) (h(6+h))/h` 

` =lim_(h to 0) 6+h` 

` =6` 

Method 3

`f'(x) = lim_(h to 0) (f(x+h)-f(x))/h` 

` =lim_(h to 0) ((x+h)^2-x^2)/h` 

` =lim_(h to 0) (x^2+2xh+h^2-x^2)/h` 

` =lim_(h to 0) (2xh+h^2)/h` 

` =lim_(h to 0) (h(2x+h))/h` 

` =lim_(h to 0) 2x+h` 

` =2x` 

So, `f'(3) = 2*3=6` 

What method would you rather do?

The students always ask "How do I decide what method to use?"

My answer has always been "If you want the derivative at just one known point then use the first definition. If you will want to know it at many points (or you don't know what point you want yet) then use the second definition."

But honestly, I would rather use the first definition all the time.  So let's.

Definition

Given a function `f(x)`  we define the derivative of `f(x)`  at `x=a`  as

`f'(a) = lim_(x to a) (f(x)-f(a))/(x-a)` 

Definition

Given a function `f(x)`  we define the derivative of `f(x)`  as

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

EG

`f(x) = x^2` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

`f'(x) = lim_(a to x) (x^2-a^2)/(x-a)` 

`f'(x) = lim_(a to x) ((x-a)(x+a))/(x-a)` 

`f'(x) = lim_(a to x) (x+a)` 

`f'(x) = x+x=2x` 

EG

`f(x) = x^3` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

`f'(x) = lim_(a to x) (x^3-a^3)/(x-a)` 

`f'(x) = lim_(a to x) ((x-a)(x^2+xa+a^2))/(x-a)` 

`f'(x) = lim_(a to x) (x^2+xa+a^2)` 

`f'(x) = x^2+x^2+x^2` 

`f'(x) = 3x^2` 

EG

`f(x) = x^n` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

`f'(x) = lim_(a to x) (x^n-a^n)/(x-a)` 

`f'(x) = lim_(a to x) ((x-a)(x^(n-1)+x^(n-2)a+x^(n-3)a^2+ cdots +xa^(n-2)+a^(n-1)))/(x-a)` 

`f'(x) = lim_(a to x) (x^(n-1)+x^(n-2)a+x^(n-3)a^2+ cdots +xa^(n-2)+a^(n-1))` 

`f'(x) = x^(n-1)+x^(n-2)x+x^(n-3)x^2+ cdots +x* x^(n-2)+x^(n-1)` 

`f'(x) = n x^(n-1)` 

EG

`f(x) = sqrt(x)` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

`f'(x) = lim_(a to x) (sqrt(x)-sqrt(a))/(x-a)` 

`f'(x) = lim_(a to x) (sqrt(x)-sqrt(a))/((sqrt(x)-sqrt(a))(sqrt(x)+sqrt(a)))` 

`f'(x) = lim_(a to x) (1)/(sqrt(x)+sqrt(a))` 

`f'(x) = (1)/(sqrt(x)+sqrt(x))` 

`f'(x) = (1)/(2sqrt(x))` 

EG

`f(x) = 1/x` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

`f'(x) = lim_(a to x) (1/x-1/a)/(x-a)` 

 `f'(x) = lim_(a to x) ((1/x-1/a)/(x-a))((ax)/(ax))` 

`f'(x) = lim_(a to x) (a-x)/((x-a)(ax))` 

`f'(x) = lim_(a to x) (-1(x-a))/((x-a)(ax))` 

`f'(x) = lim_(a to x) (-1)/(ax)` 

`f'(x) = (-1)/(x^2)` 

EG

`f(x) = b^x` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

 `f'(x) = lim_(a to x) (b^x-b^a)/(x-a)`   -->  Note: `f'(0) = lim_(a to 0) (b^0-b^a)/(0-a)= lim_(a to 0) (1-b^a)/(-a)` 

 `f'(x) = lim_(a to x) (b^x((1-b^(a-x))/(x-a)))` 

 `f'(x) = b^x lim_(a to x) (1-b^(a-x))/(x-a)` 

 `f'(x) = b^x lim_(theta to 0) (1-b^(theta))/(-theta)`    where `theta = a-x` 

`f'(x) = b^x f'(0)` 

EG

`f(x) = sin(x)` 

`f'(x) = lim_(a to x) (f(x)-f(a))/(x-a)` 

`f'(x) = lim_(a to x) (sin(x)-sin(a))/(x-a) ` 

`f'(x) = lim_(a to x) (2cos((x+a)/2)sin((x-a)/2))/(x-a)` 

 `f'(x) = (lim_(a to x) cos((x+a)/2))(lim_(a to x)(sin((x-a)/2))/((x-a)/2))` 

`f'(x) = cos(x)(lim_(theta to 0)(sin(theta))/(theta) )`    where `theta = (x-a)/2` 

`f'(x) = cos(x)` 

EG

`Sum(x) = f(x)+g(x)` 

`Sum'(x) = lim_(a to x) (Sum(x)-Sum(a))/(x-a)` 

`Sum'(x) = lim_(a to x) (f(x)+g(x)-(f(a)+g(a)))/(x-a) ` 

`Sum'(x) = lim_(a to x) (f(x)-f(a)+g(x)-g(a))/(x-a) ` 

`Sum'(x) = lim_(a to x) ((f(x)-f(a))/(x-a)+(g(x)-g(a))/(x-a) )` 

`Sum'(x) = lim_(a to x) (f(x)-f(a))/(x-a) +lim_(a to x) (g(x)-g(a))/(x-a) ` 

`Sum'(x) = f'(x)+g'(x)`