Formulas
(cumulative listing — formulas derived in exercises are linked to those exercises)

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`

Trigonometric Functions

`d/(dt)sin t=cos t`	`d/(dt)sec t=tan t sec t` (5.5 Ex. 1)
`d/(dt)cos t=-sin t`	`d/(dt)csc t=-cot t csc t` (5.5 Ex. 5)
`d/(dt)tan t=sec^2 t`	`d/(dt)cot t=-csc^2 t` (5.5 Ex. 4)

Inverse Trigonometric Functions

`d/(dt)sin^(-1) t=1/sqrt(1-t^2)`	`d/(dt)sec^(-1) t=1/(t sqrt(t^2-1)` (5.5 Ex. 10)
`d/(dt)cos^(-1) t=-1/sqrt(1-t^2)`      (5.5 Ex. 7)	`d/(dt)csc^(-1) t=-1/(t sqrt(t^2-1)` (5.5 Ex. 11)
`d/(dt)tan^(-1) t=1/(1+t^2)`	`d/(dt)cot^(-1) t=-1/(1+t^2)` (5.5 Ex. 9)

Constant Multiple Rule

`d/(dt)Af text[(]t text[)]=Ad/(dt)f text[(]t text[)],` where `A` is any constant.

Sum Rule

`d/(dt)[f text[(]t text[)]+g text[(]t text[)]]=d/(dt)f text[(]t text[)]+d/(dt)g text[(]t text[)]`

Product Rule

`d/(dt)[g text[(]t text[)]h text[(]t text[)]]=g text[(]t text[)]d/(dt)h text[(]t text[)]+h text[(]t text[)]d/(dt)g text[(]t text[)]`

Quotient Rule (4.7 Exercise 13)

`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

`(dy)/(dt)=1/((dt)/(dy))`

Chain Rule

`(dy)/(dt)=(dy)/(du)(du)/(dt)`

Special case of the Chain Rule

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

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

Combinations of the Chain Rule with specific function rules

`d/(dt)f text[(]t text[)]^r=rf text[(]t text[)]^(r-1)d/(dt)f text[(]t text[)],` for any real constant `r.`

`d/(dt)e^(f text[(]t text[)])=e^(f text[(]t text[)])d/(dt)f text[(]t text[)]`

`d/(dt)ln f text[(]t text[)]=1/(f text[(]t text[)])d/(dt)f text[(]t text[)]`

`d/(dt)sin f text[(]t text[)]=cos f text[(]t text[)] d/(dt)f text[(]t text[)]`

`d/(dt)cos f text[(]t text[)]=-sin f text[(]t text[)] d/(dt)f text[(]t text[)]`

`d/(dt)tan f text[(]t text[)]=sec^2f text[(]t text[)] d/(dt)f text[(]t text[)]`

`d/(dt)sin^(-1) f text[(]t text[)]=1/sqrt(1- f text[(]t text[)]^2) d/(dt)f text[(]t text[)]`

`d/(dt)tan^(-1) f text[(]t text[)]=1/(1+ f text[(]t text[)]^2) d/(dt)f text[(]t text[)]`