Implicit differentiation — technique, applications: Foundations

This page assumes nothing. Every scratch of notation the parent note used is unpacked below, in the order you need it. If a symbol on the parent page looked like magic, it is decoded here.

1. What a variable and a function even are

Picture a number line: $x$ is a dot you can slide left or right. That sliding is the whole game — everything downstream is a reaction to $x$ sliding.

Figure — Implicit differentiation — technique, applications

Look at the figure: the input dot $x$ on the bottom line feeds into a box labelled $f$ , and an output dot $y$ pops out on the left line. This picture is the reason we care about $\frac{d y}{d x}$ later — it shows $y$ is chained to $x$ , not free.

2. The graph — turning a rule into a curve

Notice in the figure: a vertical line cuts the circle in two places. That is the visual proof that a circle is not one function $y = f (x)$ — a function may return only one $y$ per $x$ . This single picture is why the parent note needed a whole new technique.

3. The symbol $Δ$ — "a change in"

Picture two nearby dots on a curve. Slide right by $Δ x$ ; the curve lifts you up by $Δ y$ . The ratio $\frac{Δ y}{Δ x}$ is the steepness of the little step between them — rise over run.

4. What a tangent line is (before we use the word)

In the figure, watch the amber secant lines: each joins our fixed white dot to a second dot further along. As the second dot slides in (the gap $Δ x$ shrinks), the secant swings until it settles onto the single white tangent line. That settling motion is the limit, and its final steepness is the derivative. See Tangent and Normal Lines.

5. The limit and the derivative $\frac{d y}{d x}$

6. Rules the parent leans on — and why each is true

7. Partial-derivative symbols $F_{x}, F_{y}$

For $F = x^{2} + y^{2} - 25$ : freeze $y$ , get $F_{x} = 2 x$ ; freeze $x$ , get $F_{y} = 2 y$ . The parent's slick formula $\frac{d y}{d x} = - \frac{F _{x}}{F _{y}} = - \frac{2 x}{2 y} = - \frac{x}{y}$ then falls out.

8. When the slope blows up — vertical tangents & $y = 0$

9. The special functions that show up

10. How it all feeds the topic

The figure below draws the same dependency map three tiers deep: build the derivative (top), the four rules + partials (middle), the payoff technique and its uses (bottom). Tier 1 is load-bearing — knock out the chain rule and everything below collapses.

For readers who prefer the raw graph, the same structure in text form:

Equipment checklist

Test yourself — cover the right side and answer out loud.

What does

y = f (x)

say in plain words?

Feed input

x

into rule

f

; it returns exactly one output

y

Why is a circle not an explicit function?

A vertical line hits it twice — two

y

-values for one

x

, which a function forbids.

What does

Δ x

mean?

A change in

x

x_{new} - x_{old}

What is a tangent line, in one phrase?

The straight line that just kisses the curve at a point, matching its direction there.

What question does

\frac{d y}{d x}

answer?

When

x

nudges by a hair, how fast does

y

change — i.e. the slope of the tangent.

Why do we take a limit instead of a plain ratio?

To measure steepness at one exact point on a bending curve, not the average across a gap.

Why is the power rule

\frac{d}{d x} x^{2} = 2 x

true, in a picture?

A square of side

x

gains two strips of area

x Δ x

when the side grows — the

2 x

Why is the product rule true, in a picture?

A rectangle

uv

gains one strip from each side changing:

u^{'} v + u v^{'}

State the chain rule for

g (y)

with

y = y (x)

g^{'} (y) \cdot \frac{d y}{d x}

— two gear speed-ratios multiplied.

What does

F_{x}

mean geometrically?

The slope of the landscape

F

as you walk east, with

y

frozen.

What extra condition makes

\frac{d y}{d x} = - \frac{F _{x}}{F _{y}}

legal?

F_{y} \neq = 0

at the point (the Implicit Function Theorem hypothesis; else you'd divide by zero).

What does

\frac{d y}{d x} = \frac{- 5}{0}

(5, 0)

on the circle mean?

A vertical tangent — slope undefined/infinite, not an error.

What are the domain and range of

arcsin x

x \in [- 1, 1]

and

y \in [- \frac{π}{2}, \frac{π}{2}]

, which forces

cos y \geq 0

State

\frac{d}{d x} arcsin x

\frac{1}{1 - x ^{2}}

State

\frac{d}{d x} ln x

\frac{1}{x}

Why does

ln

help with

x^{x}

ln

turns powers into products/sums:

ln (a^{b}) = b ln a

, so a variable exponent becomes a plain product.

Connections

Chain Rule — the toll-collecting engine every $\frac{d y}{d x}$ passes through.
Product Rule — for mixed $x y$ terms.
Derivatives of Inverse Functions — built from the implicit form $sin y = x$ on its fenced range.
Logarithmic Differentiation — why $ln$ tames variable exponents.
Implicit Function Theorem — the guarantee (needs $F_{y} \neq = 0$ ) that $y (x)$ secretly exists.
Tangent and Normal Lines — where the slope $\frac{d y}{d x}$ gets used, including vertical tangents.
Related Rates — same chain-rule logic, differentiating in time.