THE WHOLE UNIT · ONE REFERENCE

The Derivative
Cheat sheet.

The key rules, formulas and reminders from all 12 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

The Derivative from First Principles

Core rule

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h.

Tangent line: $y=f(a)+f'(a)(x-a)$ . Derivative units are output units divided by input units.

Watch for

A jump, corner or infinite slope can prevent a finite derivative. A finite secant slope is an approximation, not the limit itself.

Power Rules and Linearity

Core rule

Constants differentiate to zero; $(af+bg)'=af'+bg'$ for constants $a,b$ .

(x^p)'=px^{p-1}.

Watch for

Check the original domain and differentiability at endpoints or singularities. Linearity does not apply to multiplying two variable functions.

Product and Quotient Rules

Core rule

(uv)'=u'v+uv',

(u/v)'=\frac{u'v-uv'}{v^2}.

Watch for

The quotient denominator must be nonzero. Keep the subtraction order and square the whole denominator. Expanding or dividing first can simplify the calculation.

The Chain Rule

Core rule

(f\circ g)'(x)=f'(g(x))g'(x).

Differentiate the outer layer while retaining its input, then multiply by the inner derivative. Repeat for additional layers.

Watch for

Distinguish a product from a composition. Check domains before applying logarithmic or root formulas.

Exponential, Logarithmic and Trigonometric Derivatives

Core rule

(e^u)'=u'e^u,\quad(\ln|u|)'=u'/u,\quad(\sin u)'=u'\cos u,\quad(\cos u)'=-u'\sin u.

(\arcsin u)'=\frac{u'}{\sqrt{1-u^2}},

(\arctan u)'=\frac{u'}{1+u^2}.

Watch for

Trigonometric formulas use radians. Keep the original function's domain and every inner derivative.

Implicit and Logarithmic Differentiation

Core rule

Differentiate an implicit relation using $y=y(x)$ , collect $y'$ and solve only where the coefficient is nonzero.

(\ln|u|)'=u'/u,

(f^{-1})'(x)=1/f'(f^{-1}(x)).

Watch for

A variable exponent needs logarithmic differentiation or a suitable rewrite. An inverse must have a chosen branch, and its derivative denominator must be nonzero.

Rolle’s Theorem and the Mean Value Theorem

Core rule

Continuous on $[a,b]$ and differentiable on $(a,b)$ implies some $c\in(a,b)$ with $f'(c)=[f(b)-f(a)]/(b-a)$ . Equal endpoint values give Rolle's theorem.

Watch for

Check the entire interval for corners and discontinuities. A zero derivative at one point does not imply constancy.

Reading Graphs with Derivatives

Core rule

Critical candidates satisfy $f'=0$ or have no derivative while remaining in the domain. For a continuous function on a closed bounded interval, compare their values with both endpoints.

Watch for

Stationary does not mean extremal. $f''=0$ does not guarantee an inflection. An inflection needs a change in concavity.

Optimization from a Model

Core rule

Objective → constraints → one-variable domain → derivative candidates → boundary/global checks → original-variable interpretation.

Watch for

Do not optimize outside the feasible set. An open or unbounded domain requires a separate existence/global argument.

Related Rates and Units

Core rule

Differentiate the relation in time while all changing quantities remain variables. Substitute instantaneous data afterward.

Watch for

Include every chain factor. Use similar triangles or other constraints before differentiating. Distinguish signed velocity from positive speed.

Linear Approximation and Newton’s Method

Core rule

L(x)=f(a)+f'(a)(x-a),

x_{n+1}=x_n-f(x_n)/f'(x_n).

Watch for

A local approximation needs a nearby base point. Newton requires a nonzero derivative and can leave the domain or cycle. Verify the residual and use a bracket when available.

Indeterminate Forms and L’Hôpital’s Rule

Core rule

For an eligible $0/0$ or infinite-over-infinite quotient, a limiting derivative ratio $f'/g'$ determines $f/g$ under L’Hôpital's hypotheses.

Watch for

Differentiate numerator and denominator separately, not with the quotient rule. Transform products/powers first and recheck every repeated application.

The Derivative from First Principles

2 reference blocks

Read lesson ↗

Core rule

f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h.

Tangent line: $y=f(a)+f'(a)(x-a)$ . Derivative units are output units divided by input units.

Watch for

A jump, corner or infinite slope can prevent a finite derivative. A finite secant slope is an approximation, not the limit itself.

Power Rules and Linearity

2 reference blocks

Read lesson ↗

Core rule

Constants differentiate to zero; $(af+bg)'=af'+bg'$ for constants $a,b$ .

(x^p)'=px^{p-1}.

Watch for

Check the original domain and differentiability at endpoints or singularities. Linearity does not apply to multiplying two variable functions.

Product and Quotient Rules

2 reference blocks

Read lesson ↗

Core rule

(uv)'=u'v+uv',

(u/v)'=\frac{u'v-uv'}{v^2}.

Watch for

The quotient denominator must be nonzero. Keep the subtraction order and square the whole denominator. Expanding or dividing first can simplify the calculation.

The Chain Rule

2 reference blocks

Read lesson ↗

Core rule

(f\circ g)'(x)=f'(g(x))g'(x).

Differentiate the outer layer while retaining its input, then multiply by the inner derivative. Repeat for additional layers.

Watch for

Distinguish a product from a composition. Check domains before applying logarithmic or root formulas.

Exponential, Logarithmic and Trigonometric Derivatives

2 reference blocks

Read lesson ↗

Core rule

(e^u)'=u'e^u,\quad(\ln|u|)'=u'/u,\quad(\sin u)'=u'\cos u,\quad(\cos u)'=-u'\sin u.

(\arcsin u)'=\frac{u'}{\sqrt{1-u^2}},

(\arctan u)'=\frac{u'}{1+u^2}.

Watch for

Trigonometric formulas use radians. Keep the original function's domain and every inner derivative.

Implicit and Logarithmic Differentiation

2 reference blocks

Read lesson ↗

Core rule

Differentiate an implicit relation using $y=y(x)$ , collect $y'$ and solve only where the coefficient is nonzero.

(\ln|u|)'=u'/u,

(f^{-1})'(x)=1/f'(f^{-1}(x)).

Watch for

A variable exponent needs logarithmic differentiation or a suitable rewrite. An inverse must have a chosen branch, and its derivative denominator must be nonzero.

Rolle’s Theorem and the Mean Value Theorem

2 reference blocks

Read lesson ↗

Core rule

Continuous on $[a,b]$ and differentiable on $(a,b)$ implies some $c\in(a,b)$ with $f'(c)=[f(b)-f(a)]/(b-a)$ . Equal endpoint values give Rolle's theorem.

Watch for

Check the entire interval for corners and discontinuities. A zero derivative at one point does not imply constancy.

Reading Graphs with Derivatives

2 reference blocks

Read lesson ↗

Core rule

Critical candidates satisfy $f'=0$ or have no derivative while remaining in the domain. For a continuous function on a closed bounded interval, compare their values with both endpoints.

Watch for

Stationary does not mean extremal. $f''=0$ does not guarantee an inflection. An inflection needs a change in concavity.

Optimization from a Model

2 reference blocks

Read lesson ↗

Core rule

Objective → constraints → one-variable domain → derivative candidates → boundary/global checks → original-variable interpretation.

Watch for

Do not optimize outside the feasible set. An open or unbounded domain requires a separate existence/global argument.

Related Rates and Units

2 reference blocks

Read lesson ↗

Core rule

Differentiate the relation in time while all changing quantities remain variables. Substitute instantaneous data afterward.

Watch for

Include every chain factor. Use similar triangles or other constraints before differentiating. Distinguish signed velocity from positive speed.

Linear Approximation and Newton’s Method

2 reference blocks

Read lesson ↗

Core rule

L(x)=f(a)+f'(a)(x-a),

x_{n+1}=x_n-f(x_n)/f'(x_n).

Watch for

A local approximation needs a nearby base point. Newton requires a nonzero derivative and can leave the domain or cycle. Verify the residual and use a bracket when available.

Indeterminate Forms and L’Hôpital’s Rule

2 reference blocks

Read lesson ↗

Core rule

For an eligible $0/0$ or infinite-over-infinite quotient, a limiting derivative ratio $f'/g'$ determines $f/g$ under L’Hôpital's hypotheses.

Watch for

Differentiate numerator and denominator separately, not with the quotient rule. Transform products/powers first and recheck every repeated application.