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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# What is the glide ratio glide horizontal speed ut 600

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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 deriv tiv 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. Deriv tiv Below is a par tial Ta ble of Deri v a ti v es o f the more frequently encountered functions in standard form. function f(x) f (x) 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 √1 +x 2 tan −1(x) 79 differentia We may differentiate various combinations of functions using the following rules. dif entiate Let u(x) and v(x) be functions of x, the independent variable. function derivative C (constant) 0 du C ----dx du dv --- ----- + --dx dx du dv --- ----- − --dx dx du dudv dv ---v u ---++uv--- -----dx dxdx dx dv du du--dv ---v u dx--−u v dx ------ --- dx dx C. u . u+v u−v u.v 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. du(v) d dv(x) du(v) d = d . dx dv dx Higher order derivatives dy If y(x) is a function of x , then y’(x) = --- is also a function of x. If y’(x) is a --dx well-behaved function, then the derivative of y’(x) is : derivative dy’ d2y d dy --- = --- (--- ) = ----2 . --- -----y ” = --dx dx dx dx y ” is the second derivative of y(x) with respect to x. This is known as successive second successive n n−1 differentiation . In general : d y d dy ---------n = --- ( n−1 ) dx dx dx 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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