Formulas (cumulative listing)

Newton's Method

Starting from an approximation `t_0` to a zero of `f`, generate

${\displaystyle t_{n+1}=t_n-\frac{f\left(t_n\right)}{f\prime \left(t_n\right)},}$

for `n=0,1,2,...`.

Exponential Functions

`d/(dt)e^t=e^t,` where `e` is the natural base, `2.71828...`

`d/(dt)e^(kt)=ke^(kt),` for any constant `k.`

`d/(dt)b^t=text[(]ln btext[)]b^t,` for any constant `b.`

Power Rule

`d/(dt)t^r=rt^(r-1),` for any real constant `r.`

Natural logarithm function

`d/(dt)ln t=1/t`

Constant Multiple Rule

`d/(dt)Aftext[(]t text[)]=Ad/(dt)ftext[(]t text[)],` or `d/(dt)A u=A(du)/(dt),`where `A` is any constant.

Sum Rule

`d/(dt)[ftext[(]t text[)]+gtext[(]t text[)]]=f' text[(]t text[)]+g' text[(]t text[)],` or `d/(dt)text[(]u + vtext[)]=(du)/(dt)+(dv)/(dt)`

Product Rule

`d/(dt)[gtext[(]t text[)]htext[(]t text[)]]=gtext[(]t text[)] h' text[(]t text[)]+htext[(]t text[)] g' text[(]t text[)],`or `d/(dt)text[(]u vtext[)]=u (dv)/(dt)+v (du)/(dt)`

Quotient Rule (see Exercise 13 in Section 4.7)

`d/(dt)(gtext[(]t text[)]) / (htext[(]t text[)])=(htext[(]t text[)] g' text[(]t text[)] -gtext[(]t text[)] h' text[(]t text[)])/[htext[(]t text[)]]^2,` or `d/(dt) u/v=(v (du)/(dt) -u (dv)/(dt))/v^2`

Inverse Function Rule

`d/(dt) f^(-1) text[(]t text[)] = 1/[f'(f^(-1)text[(]t text[)])],` or `(dy)/(dt)=1/((dt)/(dy))`

Chain Rule

`d/(dt) f text[(] g text[(]t text[)] text[)] = f'text[(]g text[(]t text[)]text[)] g' text[(]t text[)],` or `(dy)/(dt)=(dy)/(du)(du)/(dt)`

Special case of the Chain Rule

`d/(dt)ftext[(]kt text[)]=kd/(du)ftext[(]u text[)],`

where `u=kt` and `k` is any constant.

Combinations of the Chain Rule with specific function rules

`d/(dt)ftext[(]t text[)]^r=rftext[(]t text[)]^(r-1) f' text[(]t text[)],` or `d/(dt)u^r=ru^(r-1)(du)/(dt),`for any real constant `r.`

`d/(dt)e^(ftext[(]t text[)])=e^(ftext[(]t text[)])d/(dt)ftext[(]t text[)],` or `d/(dt)e^u=e^u (du)/(dt)`

`d/(dt)ln ftext[(]t text[)]=1/(ftext[(]t text[)])d/(dt)ftext[(]t text[)],`or `d/(dt)ln u=1/u (du)/(dt)`