Derivatives & Differentiation Rules Practice Quiz with a Step-by-Step Interactive Lesson
Use the question set below to practice derivatives and differentiation rules with the exact skills you need for Calculus: derivative notation \(f'(x)\), \(\dfrac{dy}{dx}\), and \(\dfrac{d}{dx}[\,\cdot\,]\), the meaning of the derivative as an instantaneous rate of change and slope of the tangent line, the core rules (constant rule, power rule, sum/difference rule, constant multiple rule), plus the big three: product rule, quotient rule, and chain rule. You will also master must-know derivatives of trigonometric functions (\(\sin x\), \(\cos x\), \(\tan x\), \(\csc x\)), exponentials (\(e^x\), \(e^{x^2}\)), and logarithms (\(\ln x\), \(\ln(x^2)\), \(\ln(\sin x)\)). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks for expressions like \((3x-2)^4\), \(\cos(2x-1)\), \(\sqrt{x+1}\), and \((x^2+1)(x^3-1)\).
Answer the question set and review your mistakes at the end.
How this derivatives practice works
1. Take the practice set: answer the derivatives and differentiation rules questions below.
2. Open the lesson (optional): review derivative notation, the limit definition, and the main differentiation rules with clear examples.
3. Retry: return to the question set and apply the product rule, quotient rule, chain rule, and trig/log/exp derivative rules immediately.
What you will learn in the derivatives & differentiation rules lesson
Purpose: Build a clear, exam-ready understanding of derivatives and differentiation rules so you can compute derivatives quickly and correctly. You’ll learn derivative notation \(f'(x)\), \(\dfrac{dy}{dx}\), \(\dfrac{d}{dx}[f(x)]\), connect the derivative to the slope of a tangent line and instantaneous rate of change, and master the rules that appear most in quizzes and tests: constant rule, power rule, sum/difference, constant multiple, product rule, quotient rule, and especially the chain rule for composite functions. You’ll also practice the standard derivatives for trigonometric, exponential, and logarithmic functions, including composites like \((3x-2)^4\), \(\cos(2x-1)\), \(\sqrt{x+1}\), \(\ln(\sin x)\), and \(e^{x^2}\).
Success criteria
Read and write derivative notation: \(f'(x)\), \(\dfrac{dy}{dx}\), and \(\dfrac{d}{dx}[\,\cdot\,]\).
Use the constant rule: \(\dfrac{d}{dx}[c]=0\) and constant multiples: \(\dfrac{d}{dx}[cf]=c f'\).
Use the power rule: \(\dfrac{d}{dx}[x^n]=n x^{n-1}\) (including negative and fractional powers).
Differentiate sums and differences quickly: \((f\pm g)'=f'\pm g'\).
Apply the product rule: \((uv)'=u'v+uv'\).
Apply the quotient rule: \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\), \(v≠ 0\).
Apply the chain rule for composite functions: \((f(g(x)))'=f'(g(x))\,g'(x)\).
Composite function: a function inside another, like \(\cos(2x-1)\) or \((3x-2)^4\).
Chain rule: the rule used for derivatives of composite functions.
Product/quotient: expressions like \(x\sin x\) or \(\dfrac{x^2+1}{x}\) that require product or quotient rule (or smart rewriting).
Quick pre-check
Pre-check 1: What is the derivative of the constant function \(f(x)=7\)?
Hint: The slope of a constant function is zero everywhere.
Pre-check 2: What is \(\dfrac{d}{dx}\bigl[x^5 + 2x\bigr]\)?
Hint: Use the power rule on \(x^5\) and the derivative of \(2x\) is \(2\).
Derivative Basics
Derivative meaning, notation, and the core differentiation rules
Learning goal: Differentiate polynomials and basic combinations quickly using the constant rule, power rule, and linearity.
Key idea
The derivative measures how fast a function changes. For a graph \(y=f(x)\), the derivative \(f'(x)\) is the slope of the tangent line. Common notations mean the same thing: \[ f'(x),\quad \frac{dy}{dx},\quad \frac{d}{dx}[f(x)]. \] In calculus, the derivative can be defined using a limit (difference quotient): \[ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}, \] but most practice problems are solved using the differentiation rules below.
Rules you’ll use constantly
Constant rule: \(\dfrac{d}{dx}[c]=0\)
Power rule: \(\dfrac{d}{dx}[x^n]=n x^{n-1}\) (works for integers, fractions, and negatives)
Sum/difference: \((f\pm g)'=f'\pm g'\)
Constant multiple: \((cf)'=c f'\)
Worked example
Example: Differentiate \(f(x)=x^5+2x\).
Apply the power rule term-by-term: \[ \frac{d}{dx}[x^5]=5x^4,\qquad \frac{d}{dx}[2x]=2. \] So \[ f'(x)=5x^4+2. \]
Try it
Try it 1: What is \(\dfrac{d}{dx}[\,1+\sin(x)\,]\)?
Hint: The derivative of \(1\) is \(0\), and \((\sin x)'=\cos x\).
Try it 2: What is \(\dfrac{d}{dx}[x^{-1/2}]\)?
Hint: Use \(\dfrac{d}{dx}[x^n]=n x^{n-1}\) with \(n=-\tfrac{1}{2}\).
Summary
Differentiate term-by-term using linearity: constants, sums, and constant multiples.
The power rule works for negative and fractional exponents (where the function is defined).
Chain Rule
Chain rule for composite functions (inside-out differentiation)
Learning goal: Recognize composite functions and apply the chain rule cleanly to powers, radicals, trig, exponentials, and logs.
Key idea
A composite function has an “inside” function and an “outside” function. If \(y=f(g(x))\), then the chain rule says: \[ \frac{dy}{dx}=f'(g(x))\cdot g'(x). \] A quick workflow: identify the inside \(u=g(x)\), differentiate the outside with respect to \(u\), then multiply by \(\dfrac{du}{dx}\).
Worked example
Example: Differentiate \(y=(3x-2)^4\).
Let \(u=3x-2\). Then \(y=u^4\). \[ \frac{dy}{du}=4u^3,\qquad \frac{du}{dx}=3. \] Chain rule: \[ \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=4u^3\cdot 3=12(3x-2)^3. \]
Try it
Try it 1: What is the derivative of \(\cos(2x-1)\)?
Hint: \((\cos u)'=-\sin u\), then multiply by \(u'=(2x-1)'=2\).
Try it 2: What is the derivative of \(\sqrt{x+1}\)?
Hint: Rewrite \(\sqrt{x+1}=(x+1)^{1/2}\) and use the power rule + chain rule.
Summary
Composite function? Use the chain rule: outside derivative \(\times\) inside derivative.
Rewrite radicals and fractions as powers when it simplifies the chain rule.
Trig Derivatives
Trigonometric derivatives and common compositions
Learning goal: Memorize the core trig derivatives and combine them with the chain rule for expressions like \(\sin(2x)\) and \(\tan^2(x)\).
Core trig derivatives (memorize these)
\((\sin x)'=\cos x\)
\((\cos x)'=-\sin x\)
\((\tan x)'=\sec^2 x\)
\((\csc x)'=-\csc x\cot x\)
Worked example
Example: Differentiate \(\tan^2(x)\).
Write \(\tan^2(x)=(\tan x)^2\). Let \(u=\tan x\). Then \(y=u^2\). \[ \frac{dy}{du}=2u,\qquad \frac{du}{dx}=\sec^2 x. \] So \[ \frac{dy}{dx}=2u\cdot \sec^2 x=2\tan(x)\sec^2(x). \]
Try it
Try it 1: What is the derivative of \(\sin(2x)\)?
Hint: \((\sin u)'=\cos u\) and \(u=2x\) has derivative \(2\).
Try it 2: What is the derivative of \(\csc(x)\)?
Hint: This is a standard derivative: \((\csc x)'=-\csc x\cot x\).
Summary
Memorize trig derivatives, then apply chain rule to anything like \(\sin(2x)\) or \((\tan x)^2\).
Be careful with signs: \((\cos x)'=-\sin x\) and \((\csc x)'=-\csc x\cot x\).
Product & Quotient
Product rule and quotient rule (plus smart rewriting)
Learning goal: Differentiate products like \(x\sin x\) and quotients like \(\dfrac{x^2+1}{x}\) accurately and efficiently.
Use the product rule with \(u=x\) and \(v=\sin x\): \[ u'=1,\qquad v'=\cos x. \] Then \[ y'=u'v+uv' = 1\cdot \sin x + x\cdot \cos x = \sin x + x\cos x. \]
Try it
Try it 1: What is \(\dfrac{d}{dx}\bigl[(x^2+1)(x^3-1)\bigr]\)?
Hint: Product rule: \((uv)'=u'v+uv'\). Here \(u=x^2+1\), \(v=x^3-1\).
Try it 2: What is the derivative of \(\dfrac{x^2+1}{x}\) (for \(x≠ 0\))?
Hint: Simplify first: \(\dfrac{x^2+1}{x}=x+\dfrac{1}{x}\). Differentiate to get \(1-\dfrac{1}{x^2}\).
Summary
Use the product rule for products; use the quotient rule for quotients when rewriting isn’t simpler.
Algebra first can save time: \(\dfrac{x^2+1}{x}=x+\dfrac{1}{x}\).
Exp & Log
Exponential and logarithmic derivatives (plus chain rule composites)
Learning goal: Differentiate \(e^x\), \(\ln x\), and composites like \(e^{x^2}\) and \(\ln(\sin x)\) using the chain rule.
This is a composite: outside is \(\ln(u)\), inside is \(u=\sin x\). \[ \frac{d}{dx}[\ln u]=\frac{1}{u}\cdot \frac{du}{dx},\qquad \frac{du}{dx}=\cos x. \] So \[ \frac{d}{dx}[\ln(\sin x)]=\frac{1}{\sin x}\cdot \cos x=\cot x. \]
Try it
Try it 1: What is \(\dfrac{d}{dx}[\,e^{x^2}\,]\)?
Hint: \((e^u)'=e^u\cdot u'\) and \(u=x^2\) has derivative \(2x\).
Try it 2: What is \(\dfrac{d}{dx}\bigl(\ln(x^2)\bigr)\) for \(x≠ 0\)?
Hint: Chain rule: \(\dfrac{d}{dx}[\ln u]=\dfrac{u'}{u}\) with \(u=x^2\).
Summary
Exponentials: \((e^u)'=e^u\cdot u'\).
Logs: \((\ln u)'=\dfrac{u'}{u}\) (where \(u>0\)).
Many “hard” derivatives are just chain rule with these base formulas.
Strategy
Fast strategy: choose the right rule and avoid common mistakes
Learning goal: Build a reliable checklist: simplify, identify structure (sum/product/quotient/composite), then differentiate accurately.
Key idea
Most derivative problems become easy if you first identify the structure: sum/difference, constant multiple, power, product, quotient, or composite. When possible, rewrite to a simpler form: \[ \frac{1}{x}=x^{-1},\qquad \sqrt{x+1}=(x+1)^{1/2},\qquad \frac{x^2+1}{x}=x+\frac{1}{x}. \] Then apply the smallest set of rules needed.
Worked example (smart rewrite)
Example: Differentiate \(\dfrac{1}{x}\).
Rewrite: \[ \frac{1}{x}=x^{-1}. \] Power rule: \[ \frac{d}{dx}[x^{-1}]=-1\cdot x^{-2}=-\frac{1}{x^2}. \]
Try it 2: What is \(\dfrac{d}{dx}[\ln(\sin x)]\) (where \(\sin x>0\))?
Hint: \((\ln u)'=\dfrac{u'}{u}\) with \(u=\sin x\) and \(u'=\cos x\).
Final recap
Core rules: constants \(\to 0\), powers \(\to nx^{n-1}\), linearity for sums and constant multiples.
Chain rule: the most common rule in composite functions like \((3x-2)^4\), \(\cos(2x-1)\), \(e^{x^2}\), \(\ln(\sin x)\).
Product/quotient: use \((uv)'=u'v+uv'\) and \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\) (or simplify first).
Trig/exp/log: memorize the base derivatives, then apply chain rule when the input is not just \(x\).
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the rule you need (power rule, chain rule, product rule, quotient rule, trig, exponential, or logarithmic derivatives).
Practice set
Afgeleiden en differentieerregels practice questions with instant score
Answer all 10 questions below, then get your final score and a mistake review at the end so you know exactly what to improve.
0/10answered
Question 1Not answered
Wat is de afgeleide van \((2x + 5)^2\)?
Correct answer: C. \(4(2x+5)\)
Explanation: Pas de kettingregel toe: de afgeleide van \(u^2\) is \(2u\), en vermenigvuldig daarna met \(u'=2\), wat \(4(2x+5)\) geeft.
Question 2Not answered
Wat is de afgeleide van \(7\)?
Correct answer: A. 0
Explanation: De afgeleide van elke constante is \(0\).
Question 3Not answered
Wat is de afgeleide van \(5x\)?
Correct answer: D. 5
Explanation: Volgens de constante-regel geeft \(5\) maal de afgeleide van \(x\) (die \(1\) is) \(5\).
Question 4Not answered
Wat is de afgeleide van \(x^2\)?
Correct answer: A. \(2x\)
Explanation: Volgens de machtregel haal je de exponent naar voren: \(2x^{2-1}=2x\).
Question 5Not answered
Wat is de afgeleide van \(x^3\)?
Correct answer: B. \(3x^2\)
Explanation: Machtregel: de afgeleide van \(x^n\) is \(n x^{n-1}\); hier is \(n=3\), dus \(\frac{d}{dx}x^3 = 3x^2\).
Question 6Not answered
Wat is de afgeleide van \(2x^2 + 3\)?
Correct answer: C. \(4x\)
Explanation: Somregel: de afgeleide van \(2x^2\) is \(4x\), en de afgeleide van de constante \(+3\) is 0, dus \(\frac{d}{dx}(2x^2+3)=4x\).
Question 7Not answered
Wat is de afgeleide van \(x^2 + x\)?
Correct answer: D. \(2x+1\)
Explanation: Somregel: de afgeleide van \(x^2\) is \(2x\), de afgeleide van \(x\) is \(1\), dus \(\frac{d}{dx}(x^2+x)=2x+1\).
Question 8Not answered
Wat is de afgeleide van \((x-2)^2\)?
Correct answer: B. \(2(x-2)\)
Explanation: Kettingregel: voor \((x-2)^2\) neem je \(u = x-2\). Dan is \(d(u^2)/du = 2u\) en \(du/dx=1\), wat \(2(x-2)\) geeft.
Question 9Not answered
Wat is de afgeleide van \((2x)^3\)?
Correct answer: C. \(24x^2\)
Explanation: Kettingregel: voor \((2x)^3\) neem je \(u = 2x\). Dan is \(d(u^3)/du = 3u^2\) en \(du/dx=2\), dus de afgeleide is \(3(2x)^2\cdot2 = 24x^2\).
Question 10Not answered
Wat is de afgeleide van \(3x^2 - x\)?
Correct answer: D. \(6x-1\)
Explanation: Somregel: de afgeleide van \(3x^2\) is \(6x\), de afgeleide van \(-x\) is \(-1\), dus \(6x - 1\).
