Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Composite function

A function formed by plugging one function into another; if f and g are functions, then (f∘g)(x)=f(g(x)).

Composition notation (f ∘ g)(x)

The notation for the composite function defined by applying $g$ first and then $f$: $(f\circ g)(x)=f(g(x))$.

Inner function

In a composite f(g(x)), the inner function is g(x); it is evaluated first and then fed into the outer function.

Outer function

In a composite $f(g(x))$, the outer function is $f$; it acts on the output of the inner function $g(x)$.

Chain Rule

The differentiation rule for composites: $\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$.

“Outside evaluated at inside” idea

A Chain Rule phrasing: take the derivative of the outside function, evaluate it at the inside expression, then multiply by the derivative of the inside.

Leibniz notation Chain Rule

A Chain Rule form using rates: dy/dx=(dy/du)·(du/dx), emphasizing “a rate multiplied by a rate.”

Derivative operator (d/dx)

An operator meaning “differentiate with respect to x,” as in d/dx[f(x)].

Prime notation (y')

A common notation for the derivative dy/dx; for example, y'=dy/dx.

Correct evaluation in Chain Rule

In d/dx[f(g(x))], the outer derivative must be evaluated at g(x), not at x (i.e., f'(g(x))·g'(x), not f'(x)·g'(x)).

Power-chain situation

A composite where an expression is raised to a power, e.g., $(3x^2+1)^5$; differentiate the power as the outer layer and multiply by the derivative of the inside.

Chain Rule mistake: forgetting the inside derivative

A common error where students differentiate the outside but do not multiply by $g'(x)$, the derivative of the inside.

Chain Rule mistake: losing the inside in trig

A common error like writing $\cos(x)$ instead of $\cos(x^2)$ when differentiating $\sin(x^2)$; the trig derivative must keep the full inside expression.

Multiple-layer Chain Rule

Applying the Chain Rule more than once when there are several nested layers (e.g., $e^{\sqrt{1+x}}$).

Implicit differentiation

Differentiating both sides of an equation involving x and y without solving for y, treating y as y(x) and using the Chain Rule on y-terms.

Explicit function

A relationship where $y$ is isolated as a function of $x$, e.g., $y=x^2+3x$.

Implicit relationship

A relationship where $x$ and $y$ are mixed in an equation and $y$ is not isolated (e.g., $x^2+y^2=25$).

y depends on x (y(x))

In implicit differentiation, y is treated as a function of x, so differentiating y-expressions requires a dy/dx factor.

Derivative of $y^2$ with respect to $x$

Because $y=y(x)$, $\frac{d}{dx}[y^2]=2y\cdot \frac{dy}{dx}$ by the Chain Rule.

Reciprocal derivative relationship

The relationship dx/dy = 1/(dy/dx), used sometimes to switch perspectives between x as a function of y and y as a function of x.

Implicit differentiation workflow

Differentiate both sides w.r.t. x, include dy/dx on y-terms, gather dy/dx terms on one side, factor, and solve for dy/dx.

Collect dy/dx terms

A key step in implicit differentiation: move all terms containing dy/dx to one side so you can factor out dy/dx and isolate it.

Tangent line (general idea)

A line touching a curve at a point with slope equal to the derivative evaluated at that point.

Point-slope form

A tangent-line form: $y-y_1 = m(x-x_1)$, where $m$ is the slope at $(x_1,y_1)$.

Slope from implicit differentiation

After finding dy/dx implicitly, substitute the point’s x and y values into dy/dx to get the tangent slope at that point.

Second derivative implicitly

To find $\frac{d^2y}{dx^2}$, first compute $\frac{dy}{dx}$, then differentiate that expression again w.r.t. $x$ while still treating $y$ as $y(x)$.

Inverse function

A function that reverses another function; if f is one-to-one, then f^{-1} satisfies f(f^{-1}(x))=x and f^{-1}(f(x))=x.

One-to-one function

A function that passes the horizontal line test on an interval, ensuring it has an inverse function on that interval.

Inverse identity f(f^{-1}(x))=x

For $u=g(x)$, $\frac{d}{dx}[arcsin(u)]=\frac{u'}{\sqrt{1-u^2}}$ (and similarly for other inverse trig functions).

Inverse is not reciprocal

A warning: f^{-1}(x) is not the same as 1/f(x) except in special cases.

Inverse derivative formula

A general rule: (f^{-1})'(x)=1 / f'(f^{-1}(x)).

Point-specific inverse derivative formula

If f(a)=b (so f^{-1}(b)=a), then (f^{-1})'(b)=1/f'(a).

Condition f'(a) ≠ 0 for inverse derivative

For the inverse derivative to exist nicely at a point, $f$ must be differentiable at $a$ and have nonzero slope there; otherwise the inverse has a vertical tangent.

Reflection across y=x (inverse graphs)

The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ across $y=x$, and corresponding slopes are reciprocals.

Principal-value range

A restricted output range used to make inverse trig functions true functions (so trig functions become one-to-one on those intervals).

arcsin(x) principal range

For $y=arcsin(x)$, $y$ is restricted to $-\frac{\pi}{2} ≤ y ≤ \frac{\pi}{2}$.

arccos(x) principal range

For $y=arccos(x)$, $y$ is restricted to $0 ≤ y ≤ \pi$.

arctan(x) principal range

For $y=\arctan(x)$, y is restricted to $-\frac{\pi}{2} < y < \frac{\pi}{2}$.

Derivative of arcsin(x)

$\frac{d}{dx}[arcsin(x)]=\frac{1}{\sqrt{1-x^2}}$.

Derivative of arccos(x)

$\frac{d}{dx}[arccos(x)]=-\frac{1}{\sqrt{1-x^2}}$.

Derivative of arctan(x)

$\frac{d}{dx}[arctan(x)]=\frac{1}{1+x^2}$.

Derivative of arcsec(x)

$\frac{d}{dx}[arcsec(x)]=\frac{1}{|x|\sqrt{x^2-1}}$; the absolute value is important.

Derivative of arccsc(x)

$\frac{d}{dx}[arccsc(x)]=-\frac{1}{|x|\sqrt{x^2-1}}$; the absolute value is important.

Chain Rule with inverse trig

A method that takes $ln$ of both sides to simplify products/quotients/powers (especially when both base and exponent involve $x$), then differentiates implicitly.

Logarithmic differentiation

A method that takes ln of both sides to simplify products/quotients/powers (especially when both base and exponent involve x), then differentiates implicitly.

Log properties for simplification

Rules used in log differentiation: ln(ab)=ln a+ln b, ln(a/b)=ln a−ln b, and ln(a^k)=k ln a.

Derivative of ln(y) in log differentiation

Because y=y(x), d/dx[ln(y)]=(1/y)·(dy/dx) by the Chain Rule.

Derivative of x^x (via log differentiation)

If y=x^x, then y'=x^x(ln x + 1).

Outermost operation strategy

A problem-solving habit: identify the last operation applied (the “outside”) and differentiate it first, then multiply by the derivative of the inside.

Log mistake: ln(a+b) is not ln a + ln b

A common error in logarithmic differentiation: logarithms do not distribute over addition, so ln(a+b) ≠ ln a + ln b.