differentiate symbolic expression or function -凯发k8网页登录

find the derivative of the function sin(x^2).

syms f(x)
f(x) = sin(x^2);
df = diff(f,x)

df(x) = 2 x cos(x2)

find the value of the derivative at x = 2. convert the value to double.

df2 = df(2)

df2 = 4 cos(4)

double(df2)

ans = -2.6146

find the first derivative of this expression.

syms x t
df = diff(sin(x*t^2))

df = t2 cos(t2 x)

because you did not specify the differentiation variable, diff uses the default variable defined by symvar. for this expression, the default variable is x.

var = symvar(sin(x*t^2),1)

var = x

now, find the derivative of this expression with respect to the variable t.

df = diff(sin(x*t^2),t)

df = 2 t x cos(t2 x)

find the 4th, 5th, and 6th derivatives of t6.

syms t
d4 = diff(t^6,4)

d4 = 360 t2

d5 = diff(t^6,5)

d5 = 720 t

d6 = diff(t^6,6)

d6 = 720

find the second derivative of this expression with respect to the variable y.

syms x y
df = diff(x*cos(x*y), y, 2)

df = -x3 cos(x y)

compute the second derivative of the expression x*y. if you do not specify the differentiation variable, diff uses the variable determined by symvar. for this expression, symvar(x*y,1) returns x. therefore, diff computes the second derivative of x*y with respect to x.

syms x y
df = diff(x*y,2)

df = 0

if you use nested diff calls and do not specify the differentiation variable, diff determines the differentiation variable for each call. for example, differentiate the expression x*y by calling the diff function twice.

df = diff(diff(x*y))

df = 1

in the first call, diff differentiates x*y with respect to x, and returns y. in the second call, diff differentiates y with respect to y, and returns 1.

thus, diff(x*y,2) is equivalent to diff(x*y,x,x), and diff(diff(x*y)) is equivalent to diff(x*y,x,y).

differentiate this expression with respect to the variables x and y.

syms x y
df = diff(x*sin(x*y),x,y)

df = 2 x cos(x y)-x2 y sin(x y)

you also can compute mixed higher-order derivatives by providing all differentiation variables.

syms x y
df = diff(x*sin(x*y),x,x,x,y)

df = x2 y3 sin(x y)-6 x y2 cos(x y)-6 y sin(x y)

find the derivative of the function y=f(x)2dfdx with respect to f(x).

syms f(x) y
y = f(x)^2*diff(f(x),x);
dy = diff(y,f(x))

dy = 
2 f(x) ∂∂x f(x)

find the 2nd derivative of the function y=f(x)2dfdx with respect to f(x).

dy2 = diff(y,f(x),2)

dy2 = 
2 ∂∂x f(x)

find the mixed derivative of the function y=f(x)2dfdx with respect to f(x) and dfdx.

dy3 = diff(y,f(x),diff(f(x)))

dy3 = 2 f(x)

find the euler–lagrange equation that describes the motion of a mass-spring system. define the kinetic and potential energy of the system.

syms x(t) m k
t = m/2*diff(x(t),t)^2;
v = k/2*x(t)^2;

define the lagrangian.

l = t - v

l = 
m ∂∂t x(t)22-k x(t)22

the euler–lagrange equation is given by

evaluate the term ∂l/∂x˙.

d1 = diff(l,diff(x(t),t))

d1 = 
m ∂∂t x(t)

evaluate the second term ∂l/∂x.

d2 = diff(l,x)

d2(t) = -k x(t)

find the euler–lagrange equation of motion of the mass-spring system.

diff(d1,t) - d2 == 0

ans(t) = 
m ∂2∂t2 x(t) k x(t)=0

to evaluate derivatives with respect to vectors, you can use symbolic matrix variables. for example, find the derivatives ∂α/∂x and ∂α/∂y for the expression α=ytax, where y is a 3-by-1 vector, a is a 3-by-4 matrix, and x is a 4-by-1 vector.

create three symbolic matrix variables x, y, and a, of the appropriate sizes, and use them to define alpha.

syms x [4 1] matrix
syms y [3 1] matrix
syms a [3 4] matrix
alpha = y.'*a*x

alpha = yt a x

find the derivative of alpha with respect to the vectors x and y.

dx = diff(alpha,x)

dx = yt a

dy = diff(alpha,y)

dy = xt at

to evaluate a derivative with respect to a matrix, you can use symbolic matrix variables. for example, find the derivative ∂y/∂a for the expression y=xtax, where x is a 3-by-1 vector, and a is a 3-by-3 matrix. here, y is a scalar that is a function of the vector x and the matrix a.

create two symbolic matrix variables to represent x and a. define y.

syms x [3 1] matrix
syms a [3 3] matrix
y = x.'*a*x

y = xt a x

find the derivative of y with respect to the matrix a.

d = diff(y,a)

d = xt⊗x

the result is a kronecker tensor product between xt and x, which is a 3-by-3 matrix.

size(d)

ans = 1×2
     3     3

differentiate a symbolic matrix function with respect to its matrix argument.

find the derivative of the function t(x)=a⋅sin(b⋅x), where a is a 1-by-3 matrix, b is a 3-by-2 matrix, and x is a 2-by-1 matrix. create a, b, and x as symbolic matrix variables and t(x) as a symbolic matrix function.

syms a [1 3] matrix
syms b [3 2] matrix
syms x [2 1] matrix
syms t(x) [1 1] matrix keepargs
t(x) = a*sin(b*x)

t(x) = a sin(b x)

differentiate the function with respect to x using diff.

dt = diff(t,x)

dt(x) = a cos(b x)⊙b