Partial differentiations

Partial differentiations

1. From One Variable to Several

For $z = f(x,y)$, both $x$ and $y$ can change. A partial derivative measures rate of change when one variable moves, the other frozen. Notation:

Definition:

(treat $y_0$ as constant; similarly for $∂f/∂y$).

Geometrically: $f_x$ = slope of curve formed by cutting surface $z=f(x,y)$ with plane $y=y_0$; $f_y$ = slope with $x=x_0$ plane.

2. How to Compute — Treat the Other Variable as Constant

f(x,y)= x^2 y + y^3 + sin(x y):  f_x = 2x y + y cos(x y)  (y constant)
                                 f_y = x^2 +3y^2 + x cos(x y) (x constant)

f(x,y)= x^y  (x>0):  f_x = y x^{y-1} ,  f_y = x^y log x

u = log(x^2+y^2+z^2): u_x = 2x/(x^2+y^2+z^2) etc.

Chain rule and product/quotient work exactly as in one variable, once the frozen variable is treated as constant.

3. Higher-Order Partials and Clairaut's Theorem

• Second-order: $f_{xx}=∂/∂x(f_x)$, $f_{yy}$, $f_{xy}=∂/∂y(f_x)$, $f_{yx}=∂/∂x(f_y)$.
• Mixed partials equal when continuous: If $f_{xy}$ and $f_{yx}$ are continuous at $(x_0,y_0)$, then:

is Clairaut's (Young's) theorem — the order of differentiation does not matter. Counter-examples where equality fails have discontinuous mixed partials at the point.

• Notation: $∂²f/∂y∂x$ means first $x$ then $y$ (inside out).

4. Total Differential (preview)

If $z=f(x,y)$, small changes $dx, dy$ give:

This linear approximation is the workhorse for error estimation and the total derivative $df/dt = f_x dx/dt + f_y dy/dt$ when $x=x(t), y=y(t)$.

5. Worked Examples

(a) f(x,y)= x^3 y^2 + e^{xy}: f_x =3x^2 y^2 + y e^{xy}, f_y =2x^3 y + x e^{xy}
    f_{xy}= 6x^2 y + e^{xy}+ x y e^{xy}; f_{yx}= 6x^2 y + e^{xy}+ x y e^{xy} ⇒ equal ✓

(b) Verify Clairaut for f= x^2 sin y: f_x=2x sin y, f_y= x^2 cos y,
    f_{xy}=2x cos y, f_{yx}=2x cos y ✓

(c) f= r = √(x^2+y^2): f_x = x/r, f_y = y/r (r≠0); second: f_{xx}= y^2/r^3

(d) Homogeneous check: if f(x,y)= sin^{-1}(y/x): f_x= -y/(x√(x^2-y^2)), f_y=1/√(x^2-y^2)

(e) f(x,y,z)= x^y+y^z+z^x at (1,1,1): f_x= y x^{y-1}+z^x log z → at (1,1,1)=1+0=1

6. Exam Points

• Compute $f_x, f_y, f_{xx}, f_{xy}, f_{yx}$ and verify $f_{xy}=f_{yx}$ — 5 marks, very frequent.
• Find $∂z/∂x, ∂z/∂y$ for $z=log(x^2+y^2), z=tan^{-1}(y/x), z=x^y$ — 5 marks, frequent.
• Total differential $dz$ and approximate change — 3-5 marks.
• State Clairaut's theorem and condition — 3 marks.