Matlab詳解導(dǎo)數(shù)及偏導(dǎo)數(shù)運算PPT課件[通用]

1、Matlab詳解導(dǎo)數(shù)及偏導(dǎo)數(shù)運算1. 進一步理解導(dǎo)數(shù)概念及幾何意義；2. 學(xué)習(xí)Matlab的求導(dǎo)命令與求導(dǎo)法。學(xué)習(xí) Matlab 命令導(dǎo)數(shù)概念求一元函數(shù)的導(dǎo)數(shù)求多元函數(shù)的偏導(dǎo)數(shù)求高階導(dǎo)數(shù)或高階偏導(dǎo)數(shù)求隱函數(shù)所確定函數(shù)的導(dǎo)數(shù)與偏導(dǎo)數(shù)實驗內(nèi)容：1. 學(xué)習(xí)Matlab命令建立符號變量命令 sym 和 syms 調(diào)用格式：x=sym(x)建立符號變量 x；syms x y z建立多個符號變量 x,y,z；Matlab 求導(dǎo)命令 diff 調(diào)用格式：diff(f(x)，求 的一階導(dǎo)數(shù) ；diff(f(x),n)，diff(f(x，y), x)，求 對 x 的一階偏導(dǎo)數(shù) ；diff(函數(shù)f(x，y),變量

2、名 x,n)，求 對 x 的 n 階偏導(dǎo)數(shù) ；jacobian(f(x,y,z),g(x,y,z),h(x,y,z),x,y,z)matlab 求雅可比矩陣命令 jacobian，調(diào)用格式： 2. 導(dǎo)數(shù)的概念導(dǎo)數(shù)為函數(shù)的變化率，其幾何意義是曲線在一點處的切線斜率。1). 點導(dǎo)數(shù)是一個極限值例1 . 解：syms h; limit(exp(0+h)-exp(0)/h,h,0)ans=1 2). 導(dǎo)數(shù)的幾何意義是曲線的切線斜率畫出 在x=0處(P(0,1)的切線及若干條割線，觀察割線的變化趨勢.例2解:在曲線 上另取一點 ，則PM的方程是:即取h=3,2,1,0.1,0.01，分別作出幾條割線.h

3、=3,2,1,0.1,0.01;a=(exp(h)-1)./h;x=-1:0.1:3;plot(x,exp(x),r);hold onfor i=1:5;plot(h(i),exp(h(i),r.)plot(x,a(i)*x+1)endaxis square作出y=exp(x)在x=0處的切線y=1+xplot(x,x+1,r)從圖上看，隨著M與P越來越接近，割線PM越來越接近曲線的割線. 3. 求一元函數(shù)的導(dǎo)數(shù)例3 . 1) y=f(x)的一階導(dǎo)數(shù)解：輸入指令syms x;dy_dx=diff(sin(x)/x)得結(jié)果: dy_dx=cos(x)/x-sin(x)/x2.pretty(dy_

4、dx) cos(x) sin(x) - - - x 2 x在 matlab中，函數(shù) lnx 用 log(x)表示， log10(x) 表示 lgx。例4解：輸入指令syms x;dy_dx=diff(log(sin(x)得結(jié)果: dy_dx=cos(x)/sin(x).例5解：輸入指令syms x;dy_dx=diff(x2+2*x)20)得結(jié)果: dy_dx=20*(x2+2*x)19*(2*x+2).例6解：輸入指令syms a x;a=diff(sqrt(x2-2*x+5),cos(x2)+2*cos(2*x),4(sin(x),log(log(x)Matlab 函數(shù)可以對矩陣或向量操作

5、。a = 1/2/(x2-2*x+5)(1/2)*(2*x-2), -2*sin(x2)*x-4*sin(2*x), 4sin(x)*cos(x)*log(4), 1/x/log(x) 解：輸入命令2) 參數(shù)方程確定的函數(shù)的導(dǎo)數(shù)例7 dy_dx = sin(t)/(1-cos(t)syms a t;dx_dt=diff(a*(t-sin(t);dy_dt=diff(a*(1-cos(t); dy_dx=dy_dt/dx_dt.syms x y z;du_dx=diff(x2+y2+z2)(1/2),x)du_dy=diff(x2+y2+z2)(1/2),y) du_dz=diff(x2+y2+

6、z2)(1/2),z)a=jacobian(x2+y2+z2)(1/2),x y,z)解：輸入命令4. 求多元函數(shù)的偏導(dǎo)數(shù)例8du_dx=1/(x2+y2+z2)(1/2)*x du_dy =1/(x2+y2+z2)(1/2)*y du_dz = 1/(x2+y2+z2)(1/2)*z解：輸入命令syms x y;diff(atan(y/x),y) ans = -y/x2/(1+y2/x2)syms x y;diff(atan(y/x),x) ans = 1/x/(1+y2/x2)syms x y;Jacobian(atan(y/x),xy,x ,y) ans = -y/x2/(1+y2/x2

7、), 1/x/(1+y2/x2) xy*y/x, xy*log(x)5. 求高階導(dǎo)數(shù)或高階偏導(dǎo)數(shù)例10syms x ;diff(x2*exp(2*x),x,20)解：輸入命令ans = 99614720*exp(2*x)+20971520*x*exp(2*x)+1048576*x2*exp(2*x)例11syms x y ;dz_dx=diff(x6-3*y4+2*x2*y2,x,2)dz_dy=diff(x6-3*y4+2*x2*y2,y,2)dz_dxdy=diff(diff(x6-3*y4+2*x2*y2,x),y)解：輸入命令 dz_dx = 30*x4+4*y2 dz_dy = -3

8、6*y2+4*x2 dz_dxdy =8*x*y6. 求隱函數(shù)所確定函數(shù)的導(dǎo)數(shù)或偏導(dǎo)數(shù)例12syms x y ;df_dx=diff(log(x)+exp(-y/x)-exp(1),x)df_dy=diff(log(x)+exp(-y/x)-exp(1),y)dy_dx=-df_dx/df_dy解：df_dx = 1/x+y/x2*exp(-y/x)df_dy = -1/x*exp(-y/x)dy_dx = -(-1/x-y/x2*exp(-y/x)*x/exp(-y/x)例13syms x y z;a=jacobian(sin(x*y)+cos(y*z)+tan(x*z),x,y,z)dz_dx=-a(1)/a(3)dz_dy=-a(2)/a(3)解：a = cos(x*y)*y+(1+tan(x*z)2)*z, cos(x*y)*x-sin(y*z)*z, -sin(y*z)*y+(1+tan(x*z)2)*xdz_dx =(-cos(x*y)*y-(1+tan(x*z)2)*z)/(-sin(y*z)*y+(1+tan(x*z)2)*x)dz_dy =(