pkg load symbolic
symbols
x = sym("x");
t = sym("t");
f = (x + 2)*(x^2 + 3)
der1 = differentiate(f,x)
f = (t^2 + 3)*(t^(1/2) + t^3)
der2 = differentiate(f,t)
f = (x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2)
der3 = differentiate(f,x)
f = (2*x^2 + 3*x)/(x^3 + 1)
der4 = differentiate(f,x)
f = (x^2 + 1)^17
der5 = differentiate(f,x)
f = (t^3 + 3* t^2 + 5*t -9)^(-6)
der6 = differentiate(f,t)
以下是上述计算的Octave等效-
f =
(2.0+x)*(3.0+x^(2.0))
der1 =
3.0+x^(2.0)+(2.0)*(2.0+x)*x
f =
(t^(3.0)+sqrt(t))*(3.0+t^(2.0))
der2 =
(2.0)*(t^(3.0)+sqrt(t))*t+((3.0)*t^(2.0)+(0.5)*t^(-0.5))*(3.0+t^(2.0))
f =
(1.0+x^(2.0)-(2.0)*x)*(2.0-(5.0)*x^(2.0)+(3.0)*x^(3.0))
der3 =
(-2.0+(2.0)*x)*(2.0-(5.0)*x^(2.0)+(3.0)*x^(3.0))+((9.0)*x^(2.0)-(10.0)*x)*(1.0+x^(2.0)-(2.0)*x)
f =
(1.0+x^(3.0))^(-1)*((2.0)*x^(2.0)+(3.0)*x)
der4 =
(1.0+x^(3.0))^(-1)*(3.0+(4.0)*x)-(3.0)*(1.0+x^(3.0))^(-2)*x^(2.0)*((2.0)*x^(2.0)+(3.0)*x)
f =
(1.0+x^(2.0))^(17.0)
der5 =
(34.0)*(1.0+x^(2.0))^(16.0)*x
f =
(-9.0+(3.0)*t^(2.0)+t^(3.0)+(5.0)*t)^(-6.0)
der6 =
-(6.0)*(-9.0+(3.0)*t^(2.0)+t^(3.0)+(5.0)*t)^(-7.0)*(5.0+(3.0)*t^(2.0)+(6.0)*t)
Octave执行代码并返回以下结果-
f =
(2.0+x)*(3.0+x^(2.0))
der1 =
3.0+x^(2.0)+(2.0)*(2.0+x)*x
f =
(t^(3.0)+sqrt(t))*(3.0+t^(2.0))
der2 =
(2.0)*(t^(3.0)+sqrt(t))*t+((3.0)*t^(2.0)+(0.5)*t^(-0.5))*(3.0+t^(2.0))
f =
(1.0+x^(2.0)-(2.0)*x)*(2.0-(5.0)*x^(2.0)+(3.0)*x^(3.0))
der3 =
(-2.0+(2.0)*x)*(2.0-(5.0)*x^(2.0)+(3.0)*x^(3.0))+((9.0)*x^(2.0)-(10.0)*x)*(1.0+x^(2.0)-(2.0)*x)
f =
(1.0+x^(3.0))^(-1)*((2.0)*x^(2.0)+(3.0)*x)
der4 =
(1.0+x^(3.0))^(-1)*(3.0+(4.0)*x)-(3.0)*(1.0+x^(3.0))^(-2)*x^(2.0)*((2.0)*x^(2.0)+(3.0)*x)
f =
(1.0+x^(2.0))^(17.0)
der5 =
(34.0)*(1.0+x^(2.0))^(16.0)*x
f =
(-9.0+(3.0)*t^(2.0)+t^(3.0)+(5.0)*t)^(-6.0)
der6 =
-(6.0)*(-9.0+(3.0)*t^(2.0)+t^(3.0)+(5.0)*t)^(-7.0)*(5.0+(3.0)*t^(2.0)+(6.0)*t)
指数，对数和三角函数的导数
下表提供了常用的指数，对数和三角函数的导数-
函数
导数
c a.x
c a.x .lnc.a(ln是自然对数)
e x
e x
ln x
1/x
ln c x
1/x.ln c
x x
x x .(1 + ln x)
sin(x)
cos(x)
cos(x)
-sin(x)
tan(x)
sec 2 (x), or 1/cos 2 (x), 或 1 + tan 2 (x)
cot(x)
-csc 2 (x), or -1/sin 2 (x), 或 -(1 + cot 2 (x))
sec(x)
sec(x).tan(x)
csc(x)
-csc(x).cot(x)
实例
创建一个脚本文件并在其中键入以下代码-
syms x
y = exp(x)
diff(y)
y = x^9
diff(y)
y = sin(x)
diff(y)
y = tan(x)
diff(y)
y = cos(x)
diff(y)
y = log(x)
diff(y)
y = log10(x)
diff(y)
y = sin(x)^2
diff(y)
y = cos(3*x^2 + 2*x + 1)
diff(y)
y = exp(x)/sin(x)
diff(y)
运行文件时，MATLAB显示以下结果-
y =
exp(x)
ans =
exp(x)
y =
x^9
ans =
9*x^8
y =
sin(x)
ans =
cos(x)
y =
tan(x)
ans =
tan(x)^2 + 1
y =
cos(x)
ans =
-sin(x)
y =
log(x)
ans =
1/x
y =
log(x)/log(10)
ans =
1/(x*log(10))
y =
sin(x)^2
ans =
2*cos(x)*sin(x)
y =
cos(3*x^2 + 2*x + 1)
ans =
-sin(3*x^2 + 2*x + 1)*(6*x + 2)
y =
exp(x)/sin(x)
ans =
exp(x)/sin(x) - (exp(x)*cos(x))/sin(x)^2
以下是上述计算的Octave等效-
pkg load symbolic
symbols
x = sym("x");
y = Exp(x)
differentiate(y,x)
y = x^9
differentiate(y,x)
y = Sin(x)
differentiate(y,x)
y = Tan(x)
differentiate(y,x)
y = Cos(x)
differentiate(y,x)
y = Log(x)
differentiate(y,x)
% symbolic包不支持此功能
%y = Log10(x)
%differentiate(y,x)
y = Sin(x)^2
differentiate(y,x)
y = Cos(3*x^2 + 2*x + 1)
differentiate(y,x)
y = Exp(x)/Sin(x)
differentiate(y,x)
Octave执行代码并返回以下结果-
y =
exp(x)
ans =
exp(x)
y =
x^(9.0)
ans =
(9.0)*x^(8.0)
y =
sin(x)
ans =
cos(x)
y =
tan(x)
ans =
1+tan(x)^2
y =
cos(x)
ans =
-sin(x)
y =
log(x)
ans =
x^(-1)
y =
sin(x)^(2.0)
ans =
(2.0)*sin(x)*cos(x)
y =
cos(1.0+(2.0)*x+(3.0)*x^(2.0))
ans =
-(2.0+(6.0)*x)*sin(1.0+(2.0)*x+(3.0)*x^(2.0))
y =
sin(x)^(-1)*exp(x)
ans =
sin(x)^(-1)*exp(x)-sin(x)^(-2)*cos(x)*exp(x)
计算高阶导数
为了计算函数f的高阶导数，我们使用语法 diff(f,n) 。
让我们计算函数y = f(x)= x的二阶导数.e -3x
f = x*exp(-3*x);
diff(f, 2)
MATLAB执行代码并返回以下结果-
ans =
9*x*exp(-3*x) - 6*exp(-3*x)
以下是上述计算的Octave等效-
pkg load symbolic
symbols
x = sym("x");
f = x*Exp(-3*x);
differentiate(f, x, 2)
Octave执行代码并返回以下结果-
ans =
(9.0)*exp(-(3.0)*x)*x-(6.0)*exp(-(3.0)*x)
实例
在这个实例中，让我们解决一个问题。给定一个功能。我们将不得不找出方程是否成立。 y = f(x) = 3 sin(x) + 7 cos(5x) f" + f = -5cos(2x)
创建一个脚本文件并在其中键入以下代码-
syms x
y = 3*sin(x)+7*cos(5*x); % defining the functionlhs = diff(y,2)+y; %evaluting the lhs of the equation
rhs = -5*cos(2*x); %rhs of the equation
if(isequal(lhs,rhs))
disp('Yes, the equation holds true');
else
disp('No, the equation does not hold true');
end
disp('Value of LHS is: '), disp(lhs);
pkg load symbolic
symbols
x = sym("x");
y = 3*Sin(x)+7*Cos(5*x); %定义函数
lhs = differentiate(y, x, 2) + y; %计算方程 lhs
rhs = -5*Cos(2*x); %方程式 rhs
if(lhs == rhs)
disp('Yes, the equation holds true');
else
disp('No, the equation does not hold true');
end
disp('Value of LHS is: '), disp(lhs);
