Practice Explanation vs Derivation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The distinction between explaining WHY a result is true (conceptual insight) and showing HOW it can be derived step by step (procedural derivation).

Derivation: here are the steps. Explanation: here's why it makes sense.

Showing a random 20 of 50 problems.

Example 1

medium

Why is the derivative of

x^2

equal to

2x

? Give a geometric/algebraic explanation, not just the power rule.

Example 2

medium

Why is the median sometimes better than the mean? Explanation: it resists outliers. For data

1,2,3,4,100

, give the median.

Example 3

challenge

Why does Gaussian elimination work? Explanation: row operations preserve the solution set. Solving

\begin{cases}x+y=5\\x-y=1\end{cases}

, give

x

Example 4

easy

Why is

a^0 = 1

for nonzero

a

? Give a pattern-based explanation, not a derivation.

Example 5

medium

The derivation of the derivative of

x^2

uses limits; the explanation is 'slope of the tangent.' Give

\frac{d}{dx}x^2

x=3

Example 6

medium

The Pythagorean theorem is

a^2 + b^2 = c^2

. Give a one-sentence explanation of why squares appear, not just a derivation.

Example 7

medium

A short derivation can hide insight. The product rule

(fg)'=f'g+fg'

is explained by area-change of a rectangle. If

f=g=x

x=2

, give

(fg)'=(x^2)'

there.

Example 8

easy

A formula

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

is derived from Pythagoras. For points

(0,0)

and

(3,4)

, give

d

Example 9

medium

The derivation of

1+2+\cdots+n=\frac{n(n+1)}{2}

uses pairing; the explanation is a staircase doubling into a rectangle. For

n=6

, give the rectangle's area (which is

n(n+1)

Example 10

easy

The derivation of

(a+b)^2=a^2+2ab+b^2

uses FOIL. The explanation is a square split into regions. How many regions does the geometric picture have? Give the count.

Example 11

easy

The derivation of the circle area uses integration; the explanation unrolls it into a triangle of base

2\pi r

and height

r

. What is that triangle's area?

Example 12

challenge

Derive

e^{i\pi} + 1 = 0

from Euler's formula. Explain its conceptual content beyond the algebraic manipulation.

Example 13

hard

The derivative of

\sin x

\cos x

. Explain why this makes geometric sense in terms of the unit circle.

Example 14

medium

Why does the chain rule

(f \circ g)' = f'(g) \cdot g'

multiply two derivatives? Explain conceptually.

Example 15

easy

You know that

(a+b)^2 = a^2+2ab+b^2

. Give an explanation (not just algebraic expansion) of why the cross-term

2ab

appears.

Example 16

easy

A derivation shows

0.999\ldots=1

10x-x=9

. The explanation: no number fits between them. What is

9x

x=0.999\ldots

Example 17

medium

Derive

\binom{n}{k} = \frac{n!}{k!(n-k)!}

from the definition. Then explain WHY the symmetry

\binom{n}{k} = \binom{n}{n-k}

holds.

Example 18

hard

Derive Euler's formula

e^{i\theta} = \cos\theta + i\sin\theta

via Taylor series. Then give a one-sentence explanation involving rotation.

Example 19

challenge

Two proofs that

\sqrt 2

is irrational: a parity-based contradiction and a unique-factorization argument. Which one offers a deeper EXPLANATION, and why?

Example 20

easy

The area of a triangle is

\tfrac{1}{2} b h

. Give the geometric explanation (NOT a formal derivation) for the

\tfrac{1}{2}

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Related Concepts

Proof (Intuition)