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

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

Df(x) = $$2\hspace{0.17em}x\hspace{0.17em}\cos \left(x^2\right)$$

Find the value of the derivative at x = 2. Convert the value to double.

Df2 = Df(2)

Df2 = $$4\hspace{0.17em}\cos \left(4\right)$$

double(Df2)

ans = -2.6146

Find the first derivative of this expression.

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

Df = $$t^2\hspace{0.17em}\cos \left(t^2\hspace{0.17em}x\right)$$

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 = $$2\hspace{0.17em}t\hspace{0.17em}x\hspace{0.17em}\cos \left(t^2\hspace{0.17em}x\right)$$

Find the 4th, 5th, and 6th derivatives of $$t^6$$.

syms t
D4 = diff(t^6,4)

D4 = $$360\hspace{0.17em}{t}^2$$

D5 = diff(t^6,5)

D5 = $$720\hspace{0.17em}t$$

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 = $$-x^3\hspace{0.17em}\cos \left(x\hspace{0.17em}y\right)$$

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 = $$2\hspace{0.17em}x\hspace{0.17em}\cos \left(x\hspace{0.17em}y\right)-x^2\hspace{0.17em}y\hspace{0.17em}\sin \left(x\hspace{0.17em}y\right)$$

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 = $$x^2\hspace{0.17em}{y}^3\hspace{0.17em}\sin \left(x\hspace{0.17em}y\right)-6\hspace{0.17em}x\hspace{0.17em}{y}^2\hspace{0.17em}\cos \left(x\hspace{0.17em}y\right)-6\hspace{0.17em}y\hspace{0.17em}\sin \left(x\hspace{0.17em}y\right)$$

Find the derivative of the function $$y=f(x{)}^2\frac{df}{dx}$$ with respect to $$f(x)$$

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

Dy = 
$$2\hspace{0.17em}f\left(x\right)\hspace{0.17em}\frac{\partial }{\partial x}\mathrm{ }f\left(x\right)$$

Find the 2nd derivative of the function $$y=f(x{)}^2\frac{df}{dx}$$ with respect to $$f(x)$$

Dy2 = diff(y,f(x),2)

Dy2 = 
$$2\hspace{0.17em}\frac{\partial }{\partial x}\mathrm{ }f\left(x\right)$$

Find the mixed derivative of the function $$y=f(x{)}^2\frac{df}{dx}$$ with respect to $$f(x)$$ and $$\frac{df}{dx}$$.

Dy3 = diff(y,f(x),diff(f(x)))

Dy3 = $$2\hspace{0.17em}f\left(x\right)$$

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 = 
$$\frac{m\hspace{0.17em}{\left(\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\right)}^2}{2}-\frac{k\hspace{0.17em}{x\left(t\right)}^2}{2}$$

The Euler–Lagrange equation is given by

Evaluate the term $$\partial L/\partial \underset{}{\overset{\dot{}}{x}}$$.

D1 = diff(L,diff(x(t),t))

D1 = 
$$m\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)$$

Evaluate the second term $$\partial L/\partial x$$.

D2 = diff(L,x)

D2(t) = $$-k\hspace{0.17em}x\left(t\right)$$

Find the Euler–Lagrange equation of motion of the mass-spring system.

diff(D1,t) - D2 == 0

ans(t) = 
$$m\hspace{0.17em}\frac{{\partial }^2}{\partial t^2}\mathrm{ }x\left(t\right)+k\hspace{0.17em}x\left(t\right)=0$$