MTH251

Notes 3.3

Fall 2018

Patterns

1. `f'(x)=d/(dx)[x] = lim_{h to 0} ((x+h)-x)/(h)=1`
2. `f'(x)=d/(dx)[x^2] = lim_{h to 0} ((x+h)^2-x^2)/(h) = lim_{h to 0} (2x+h) = 2x`
3. `f'(x)=d/(dx)[x^3] = lim_{b to x} (x^3-b^3)/(x-b) = lim_{b to x} (x^2+bx+b^2) = 3x^2`
4. `f'(x)=d/(dx)[1/x] = lim_{b to x} (1/x-1/b)/(x-b) = lim_{b to x} (b-x)/(xb)*1/(x-b) = lim_{b to x} (-1)/(xb) = (-1)/x^2 = -x^-2`
5. `f'(x)=d/(dx)[sqrt(x)] = lim_{b to x} (sqrt(x)-sqrt(b))/(x-b) = lim_{b to x} 1/(sqrt(x)+sqrt(b)) = 1/(2sqrt(x))=1/2x^(-1/2)`
6. What is the pattern?

General Rules for the derivative:

The Easy Ones

1. `d/(dx)[a*f(x)] = a*d/(dx)[f(x)]`
2. `d/(dx)[f(x)+g(x)] = d/(dx)[f(x)]+d/(dx)[g(x)] = f'(x)+g'(x)`

The Harder Ones

1. `d/(dx)[f(x)*g(x)] = ?`
2. `d/(dx)[f(x)/g(x)] = ?`

`d/(dx)[f(x)*g(x)] = lim_(h->0) (f(x+h)*g(x+h)-f(x)*g(x))/h`

The trick is to add and subtract `f(x+h)*g(x)` in the numerator.

`lim_(h->0) (f(x+h)*g(x+h)-f(x)*g(x))/h = lim_(h->0) (f(x+h)*g(x+h)-f(x+h)*g(x)+f(x+h)*g(x)-f(x)*g(x))/h`

Now, factor in groups.

`lim_(h->0) (f(x+h)*g(x+h)-f(x+h)*g(x)+f(x+h)*g(x)-f(x)*g(x))/h = (f(x+h) [color(blue)(g(x+h)-g(x))]+g(x)[color(blue)(f(x+h)-f(x))])/h`

Now separate these two groups.

`lim_(h->0) (f(x+h)[g(x+h)-g(x)]+g(x)[f(x+h)-f(x)])/h = lim_(h->0) (f(x+h)[g(x+h)-g(x)])/h + lim_(h->0) (g(x)[f(x+h)-f(x)])/h`

We can evaluate these limits based on our limit rules.

`f(x)*g'(x)+g(x)*f'(x)`