Derivatives & Differentiation Rules Practice Questions, Quiz, and Step-by-Step Lesson - improve your math skills with focused questions and clear explanations.
Derivatives & Differentiation Rules Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page 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 \((x^2+1)(x^3-1)\).
How this derivatives practice works
1. Take the quiz: answer the derivatives and differentiation rules questions at the top of the page.
2. Open the lesson (optional): review derivative notation, the limit definition, and the main differentiation rules with clear examples.
3. Retry: return to the quiz 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
Exponential derivatives: \((e^x)'=e^x\), \((ae^x)'=ae^x\), and chain rule for \(e^{x^2}\)
Log derivatives: \((\ln x)'=\dfrac{1}{x}\); chain rule for \(\ln(x^2)\) and \(\ln(\sin x)\)
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing derivatives and differentiation rules.
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Derivatives & Rules
Step-by-step guide
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Derivatives & Differentiation Rules Lesson
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Lesson Overview
Lesson overview
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).
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