2014-AP微积分CALCULUS知识点总结
AP CALCULUS Derivative
A DERIVATIVE FUNCTION
1. The derivative function or simply the derivative is defined as
f?(x)＝y?＝lim?x?02. Find the derivative function a) Find ?y,
?yf(x??x)?f(x)
?lim?x?x?0?xb) Find the average rate of change ?y, ?xc) Find the limit lim?y.
?x?0?x3. Geometric significance
Consider a general function y=f(x), a fixed point A(a,f(a)) and a variable point B(x,f(x)). The slope of chord AB=f(x)?f(a).
x?aNow as B?A, x?a and the slope of chord AB?slope of tangent at A. So, limf(x)?f(a) is f?(a).
x?ax?aThus, we can know the derivative at x=a is the slope of the tangent at x=a.
4. Rules f(x) f?(x) C(a constant) 0 xn sinx cosx tanx nxn?1 cosx ?sinx sec2x?1 cos2xarcsinx 1 1-x2 2
AP CALCULUS Derivative
lnx 1x logax ex ax u(x)?v(x) 1logae xex axlna u?(x)?v?(x) u(x)v(x) u(x) v(x) u?(x)v(x)?u(x)v?(x) u?v?uv? (v?0) 2v
5. The chain rule
If y?f(u) where u?u(x) then
dydydu?. dxdudxf(x)?eg(x) f?(x)?eg(x)g?(x)
g?(x) f(x)?lng(x) f?(x)?g(x)f(x)?u(x)v(x)?elnu(x)v(x)?ev(x)lnu(x),
f?(x)?ev(x)lnu(x)v(x)u?(x)[v?(x)lnu(x)?]
u(x)6. Inverse function, Parametric function and Implicit function Inverse function:
dy11?, f?(x)??1,
[f(x)]?dxdxdyi.e., y?arcsinx, x?siny
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2014AP微积分CALCULUS知识点总结
