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Derivatives, Chain Rule & Implicit Change

Useful preparation: Limits & Continuity

For a long equation, swipe sideways inside its formula box to see every term.

Meaning & notation

A derivative describes a function's local linear behavior. Shrinking an input interval turns average change into instantaneous change. Nested functions transmit change through layers, while implicit equations couple the changes of their variables.

Derivative: The finite limit of the difference quotient at an input. It gives the tangent slope and the instantaneous output change per unit input.
Linear approximation: The tangent-line estimate near a chosen input. Differentiability means the approximation error becomes negligible relative to the input increment.
Implicit differentiation: Differentiating an equation while treating one variable as a local function of another, so every occurrence of that dependent variable carries its own derivative.

Conditions before calculation

For an ordinary two-sided derivative, the function must be defined on a neighborhood of the input and the difference-quotient limit must be finite.
The chain rule requires the inner function to be differentiable at the input and the outer function to be differentiable at the resulting inner value.
For a continuously differentiable implicit equation, a nonzero partial derivative with respect to the dependent variable guarantees a local differentiable branch; zero requires separate analysis.

Why the method works

1. Start from the local rate

The difference quotient compares output change with input change. Its limit gives the tangent slope and local linear coefficient; h denotes the increment.

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h,\qquad f(a+h)=f(a)+f'(a)h+o(h)

2. Multiply the rates through a composition

An input change first changes g, then that intermediate change changes f. Their local conversion factors multiply.

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

3. Differentiate each dependency

Along a solution branch, F remains zero, so its total rate vanishes. Solve for the unknown slope when its coefficient is nonzero.

F(x,y(x))=0\ \Longrightarrow\ F_x+F_y y'=0,\qquad y'=-\frac{F_x}{F_y}\quad(F_y\ne0)

Worked examples

EXAMPLE 1

Differentiate the nested polynomial and find its tangent slope at input 1.

Name the inner expression and differentiate the outer cube with respect to that expression.
$y=(1+2x^2)^3,\qquad u=1+2x^2,\qquad \frac{dy}{du}=3u^2$
Differentiate the inner expression with respect to x and multiply the two rates.
$\frac{du}{dx}=4x,\qquad y'=3(1+2x^2)^2(4x)=12x(1+2x^2)^2$
Evaluate only after keeping both factors. The point value also supplies the tangent-line anchor.
$y(1)=27,\qquad y'(1)=108,\qquad y-27=108(x-1)$

Conclusion: The tangent slope is 108; omitting the inner derivative would give the wrong local rate.

EXAMPLE 2

Find the tangent to the implicit curve at the point with coordinates 1 and 2.

Verify the point lies on the curve. The partial derivative with respect to y is nonzero there, so a local y branch is justified.
$x^2+xy+y^2=7,\qquad 1+2+4=7,\qquad F_y(1,2)=1+4=5$
Use the product rule on the mixed term and the chain rule on the squared dependent variable.
$2x+y+xy'+2yy'=0$
Collect the dependent-variable derivatives before dividing, then evaluate at the stated point.
$y'=-\frac{2x+y}{x+2y},\qquad y'(1)=-\frac45,\qquad y-2=-\frac45(x-1)$

Conclusion: The negative slope describes the local branch without solving the entire curve explicitly.

Pitfalls & counterexamples

Tempting claim: Every continuous function has a derivative.

Correction: Continuity controls function values; differentiability additionally requires a consistent finite local slope.

Test the claim: The absolute-value function is continuous at zero, but its left slope is negative one and its right slope is positive one.

Check your understanding

A derivative has units of meters per second. What units can its input and output have?

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Compare your reasoning

Time in seconds and position in meters.

The derivative's units are output units divided by input units, matching the difference quotient.

Explore an interactive illustration

A secant becomes a tangent

Parent curve $\displaystyle y = x^{2}$ . Fix A at $\displaystyle x = a$ . Bring B closer to A to explore an instantaneous rate.

\displaystyle a = 1

Distance

\displaystyle h = 1

Watch the secant approach the tangent

Red secant: $\displaystyle \frac{\left(a + h\right)^{2} - a^{2}}{h} = 2\, a + h = 3$ . Dashed green tangent: limit as $\displaystyle h \to 0 = 2\, a = 2$ . The limit avoids dividing by zero: simplify for $\displaystyle h \ne 0$ , then let h approach zero.

Slope gap: 1 · The animation approaches $\displaystyle h = 0$ ; it never divides by zero.

Connect this idea

Derivatives lead to optimization and differential equations. The same dependency bookkeeping becomes the multivariable chain rule and gradient.

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Reference reading

Use these texts to study the underlying definitions and theorems in greater depth. The explanations and examples above are written for Numerive.