Partial derivatives — notation, calculation, geometric meaning

WHY do we need partial derivatives?

In single-variable calculus, $f(x)$ has one slope $f'(x)$ . But a surface like $z=f(x,y)$ has infinitely many directions you could move at a point. There is no single "slope". So we ask a simpler, more honest question:

If I only walk in the $x$ -direction (keeping $y$ fixed), how fast does the height change?

That gives the partial derivative with respect to $x$ . Same idea for $y$ . These two special directions are the foundation everything else (gradient, directional derivative, tangent plane) is built from.

WHAT is a partial derivative? (definition from first principles)

WHY this is just an ordinary derivative: define a single-variable function $g(x)=f(x,b)$ (freeze $y=b$ ). Then $\frac{\partial f}{\partial x}(a,b)=g'(a).$ So all your old differentiation rules still apply — you just treat the frozen variable as a number.

Notation (all mean the same thing)

Evaluated at a point: $\left.\dfrac{\partial f}{\partial x}\right|_{(a,b)}$ or $f_x(a,b)$ .

HOW to calculate — the freeze-and-differentiate algorithm

Pick the variable you differentiate with respect to.
Treat every other variable as a constant (a fixed number like $7$ ).
Differentiate normally using all the usual rules (power, product, chain...).
Plug in the point if a number is asked.

GEOMETRIC meaning — slices of the surface

Figure — Partial derivatives — notation, calculation, geometric meaning

These two slopes are the two tangent lines spanning the tangent plane at $(a,b)$ : $z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b).$ WHY: the tangent plane must reproduce both slice-slopes, so the coefficients are exactly the partials.

Higher-order & mixed partials (a peek)

You can differentiate again: $f_{xx}=\partial_x(f_x)$ , and mixed $f_{xy}=\partial_y(f_x)$ .

Common mistakes (steel-manned)

Active Recall

Recall Forecast then verify

Q: For $f(x,y)=e^{x}\cos y$ , predict $f_x$ and $f_y$ before reading. A: $f_x=e^x\cos y$ (cos y is constant), $f_y=-e^x\sin y$ ( $e^x$ constant).

Recall Feynman: explain to a 12-year-old

Imagine a hilly field. You're standing at a spot. If you take ONE step straight East and see how much higher or lower you got, that's the "East slope" — the partial derivative in $x$ . Take one step North instead → that's the "North slope", the partial in $y$ . You measure each direction separately by pretending you can only walk that one way. The hill doesn't change; you just choose which path to test.

Flashcards

What does

\partial f/\partial x

measure?

Rate of change of

f

x

varies with all other variables held constant.

Limit definition of

f_x(a,b)

\lim_{h\to0}\frac{f(a+h,b)-f(a,b)}{h}

How do you compute a partial derivative practically?

Treat all other variables as constants and differentiate normally.

Geometric meaning of

f_x(a,b)

Slope of the tangent line to the curve cut by the plane

y=b

x=a

Why use

\partial

instead of

d

Because

f

depends on several variables;

\partial

signals others are held fixed.

f=x^2y

: find

f_x

2xy

(not

2x

— keep the

y

f=\sin(xy^2)

: find

f_y

\cos(xy^2)\cdot 2xy

Clairaut's theorem states?

If second partials are continuous,

f_{xy}=f_{yx}

Tangent plane formula at

(a,b)

z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)

Common error: differentiating

-7y

f_x

It's a constant w.r.t.

x

, so its

x

-derivative is

0

Connections

Gradient vector — packages $f_x,f_y$ into $\nabla f$ .
Directional derivative — generalizes partials to any direction.
Tangent plane and linear approximation — built from the two partials.
Chain rule (multivariable) — partials chain together.
Total differential — $df=f_x\,dx+f_y\,dy$ .
Single-variable derivative — the special case partials reduce to.
Clairaut's theorem — symmetry of mixed partials.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, single-variable calculus mein function $f(x)$ ka ek hi slope hota hai. Lekin jab function do (ya zyada) variables ka ho, jaise $z=f(x,y)$ , toh ek hi point pe bohot saari directions mein chal sakte ho — toh "ek slope" ka koi matlab nahi banta. Isliye hum simple sawaal poochte hain: "Sirf $x$ ki taraf chaloon, $y$ ko freeze karke, toh height kitni fast badlegi?" Yahi hai partial derivative $\partial f/\partial x$ .

Calculation ka funda ekdum easy hai: jis variable ke respect mein differentiate kar rahe ho usko chhodke baaki sabko constant (number) maan lo, aur phir normal differentiation rules laga do. Jaise $f=x^2y$ mein $f_x = 2xy$ — yahan $y$ ko constant coefficient ki tarah saath le jaate hain, usse drop nahi karte. Aur jis term mein $x$ hai hi nahi, uska $x$ -derivative seedha $0$ .

Geometric meaning samjho: surface ko ek vertical plane se kaato. Agar $y=b$ fix karo, toh ek curve milta hai, aur $f_x(a,b)$ us curve ki tangent line ka slope hai. Isi tarah $x=a$ fix karke $f_y$ milta hai. Ye do slopes milke tangent plane banate hain — pure multivariable calculus ki neev yahi hai (gradient, directional derivative, optimization sab isi pe khade hain).

Yaad rakhne ka mantra: "Freeze the rest, slope the best." Aur do galtiyan mat karna — (1) doosre variable ko bhoolna mat, woh constant ban ke saath rehta hai; (2) point ki value pehle mat daalo, pehle differentiate karo, phir number plug karo.

Partial derivatives — notation, calculation, geometric meaning

WHY do we need partial derivatives?

WHAT is a partial derivative? (definition from first principles)

Notation (all mean the same thing)

HOW to calculate — the freeze-and-differentiate algorithm

GEOMETRIC meaning — slices of the surface

Higher-order & mixed partials (a peek)

Common mistakes (steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Multivariable Calculus

Connections