1、實(shí)驗(yàn)3 導(dǎo)數(shù)及偏導(dǎo)數(shù)運(yùn)算,實(shí)驗(yàn)?zāi)康模?1. 進(jìn)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(shí)驗(yàn)內(nèi)容：,1. 學(xué)習(xí)Matlab命令,建立符號(hào)變量命令 sym 和 syms 調(diào)用格式：,x=sym(x),建立符號(hào)變量 x；,syms x y z,建立多個(gè)符號(hào)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(duì) x 的

2、一階偏導(dǎo)數(shù) ；,diff(函數(shù)f(x，y),變量名 x,n)，,求 對(duì) 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(diǎn)處的切線斜率。,1). 點(diǎn)導(dǎo)數(shù)是一個(gè)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(shì).,例2,解:在曲線 上另取一點(diǎn) ，

3、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=3,2,1,0.1,0.01;a=(exp(h)-1)./h;x=-1:0.1:3; plot(x,exp(x),r);hold on for i=1:5; plot(h(i),exp(h(i),r.) plot(x,a(i)*x+1) end axis square,作出y=exp(x)在x=0處的切線y=1+x,plot(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

4、=diff(sin(x)/x),得結(jié)果: dy_dx=cos(x)/x-sin(x)/x2.,pretty(dy_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=di

5、ff(sqrt(x2-2*x+5),cos(x2)+2*cos(2*x), 4(sin(x),log(log(x),Matlab 函數(shù)可以對(duì)矩陣或向量操作。,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.,s

6、yms 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+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8,du_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

7、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), 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10,syms 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11,syms x y ; dz_dx=diff(x6-3*y4+2*x2*y2,x,

8、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 = -36*y2+4*x2 dz_dxdy =8*x*y,6. 求隱函數(shù)所確定函數(shù)的導(dǎo)數(shù)或偏導(dǎo)數(shù),例12,syms 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_d

9、y = -1/x*exp(-y/x) dy_dx = -(-1/x-y/x2*exp(-y/x)*x/exp(-y/x),例13,syms 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)*x dz_dx = (-cos(x*y)*y-(1+tan(x*z)2)*z)/(-sin(y*z)*y+(1+tan(x*z)2)*x) dz_dy =