For the function fx y y tan1 xy1 x2 yfind f xySolutionfxy

For the function f(x, y) = y tan^-1 xy/1 + x^2 y^find f xy.

Solution

fxy is calculated as follows

fxy = 2f / yx = (f / x) / y

syms x

syms y

f = (y(tan^-1(x*y))/(1+(x^2*y^2)));

derx = differentiate(f,x);

dery = differentiate(derx,y);

Following are the rules for differentiation

Rule 1

For any functions f and g and any real numbers a and b are the derivative of the function:

h(x) = af(x) + bg(x) with respect to x is given by

h\'(x) = af\'(x) + bg\'(x)

Rule 2

The sum and subtraction rules state that if f and g are two functions, f\' and g\' are their derivatives respectively, then,

(f + g)\' = f\' + g\'

(f - g)\' = f\' - g\'

Rule 3

The product rule states that if f and g are two functions, f\' and g\' are their derivatives respectively, then,

(f.g)\' = f\'.g + g\'.f

Rule 4

The quotient rule states that if f and g are two functions, f\' and g\' are their derivatives respectively, then,

(f/g)\' = (f\'.g - g\'.f)/g2

Rule 5

The polynomial or elementary power rule states that, if y = f(x) = xn, then f\' = n. x(n-1)

A direct outcome of this rule is that the derivative of any constant is zero, i.e., if y = k, any constant, then

f\' = 0

Rule 6

The chain rule states that, derivative of the function of a function h(x) = f(g(x)) with respect to x is,

h\'(x)= f\'(g(x)).g\'(x)

Example

Create a script file and type the following code into it

syms x

syms t

f = (x + 2)*(x^2 + 3)

der1 = diff(f)

f = (t^2 + 3)*(sqrt(t) + t^3)

der2 = diff(f)

f = (x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2)

der3 = diff(f)

f = (2*x^2 + 3*x)/(x^3 + 1)

der4 = diff(f)

f = (x^2 + 1)^17

der5 = diff(f)

f = (t^3 + 3* t^2 + 5*t -9)^(-6)

der6 = diff(f)

When you run the file, MATLAB displays the following result

f =

(x^2 + 3)*(x + 2)

der1 =

2*x*(x + 2) + x^2 + 3

f =

(t^(1/2) + t^3)*(t^2 + 3)

der2 =

(t^2 + 3)*(3*t^2 + 1/(2*t^(1/2))) + 2*t*(t^(1/2) + t^3)

f =

(x^2 - 2*x + 1)*(3*x^3 - 5*x^2 + 2)

der3 =

(2*x - 2)*(3*x^3 - 5*x^2 + 2) - (- 9*x^2 + 10*x)*(x^2 - 2*x + 1)

f =

(2*x^2 + 3*x)/(x^3 + 1)

der4 =

(4*x + 3)/(x^3 + 1) - (3*x^2*(2*x^2 + 3*x))/(x^3 + 1)^2

f =

(x^2 + 1)^17

der5 =

34*x*(x^2 + 1)^16

f =

1/(t^3 + 3*t^2 + 5*t - 9)^6

der6 =

-(6*(3*t^2 + 6*t + 5))/(t^3 + 3*t^2 + 5*t - 9)^7