Unformatted text preview: n a
capacitor, cur rent, voltage, resistance, energy, cost, population, time,
probability, or any other quantity that CHANGES. The calculation to find the
deri v a ti v e f ’ (x) , the expression of the INSTANTANEOUS RATE OF CHANGE,
from the function f(x), the expression of CHANGE, is almost mechanical.
Below is a par tial Ta ble of Deri v a ti v es o f the more frequently encountered
functions in standard form.
derivative f ’(x)
xn (n is a rational number) n xn−1 ex ex log (x) 1/ x sin (x) cos (x) cos (x) − sin (x) tan (x) sec2 (x) cosec (x) − cosec (x). cot (x)
. sec (x) sec (x) . tan (x) cot (x) − cosec2 (x) sin −1(x) √1 − x 2 1 − cos −1(x) 1
√1 − x 2
√1 +x 2 tan −1(x)
We may differentiate various combinations of functions using the following rules.
Let u(x) and v(x) be functions of x, the independent variable.
function derivative C (constant) 0
--- ----- + --dx dx
--- ----- − --dx dx
---v u ---++uv--- -----dx
---v u dx--−u v dx
------ --- dx
dx C. u
/v v2 Chain r ule
Let u(v) be a function of v and v(x) be a function of x, the independent variable.
= d . dx
Higher order derivatives
If y(x) is a function of x , then y’(x) = --- is also a function of x. If y’(x) is a
well-behaved function, then the derivative of y’(x) is :
--- = --- (--- ) = ----2 .
--- -----y ” = --dx
y ” is the second derivative of y(x) with respect to x. This is known as successive
differentiation . In general : d y
---------n = --- ( n−1 )
80 Example : y(t) is the function that expresses CHANGE in height.
y ’ (t) is the INSTANTANEOUS RATE OF CHANGE in height or ver tical speed.
We can think of y ’ (t) as the function that expresses CHANGE in ver tical
speed and y ”(t) the INSTANTANEOUS RATE OF CHANGE in ver tical s...
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- Fall '09
- Limit, Δx