📖 Definition
The Derivative: Calculus' Core Definition
The derivative is a limit — nothing more.
↳ Every derivative rule (power, product, chain) is built from this single limit definition.
Kit Library / Mathematics / Calculus
⚡ Topic Learning KitApplied Calculus — Unit 1–3 Important Questions (Part A)
English 18 leveled MCQs 22 flashcards 9 games Free
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Veda Bites
📖 Definition
➗ Formula
✏️ Example
💡 Key Idea
📖 Definition
➗ Formula
📖 Definition
📖 Definition
💡 Key Idea
⚙️ Process
➗ Formula
📖 Definition
📖 Definition
➗ Formula
⚠️ Common Mistake
🎯 Exam Tip
The whole idea, one bite at a time
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
📖 Definition
➗ Formula
Power Rule: Derivative of xⁿ
Bring the power down, subtract one.
↳ For f(x) = x², f'(x) = 2x; for f(x) = x⁴ + x², f'(x) = 4x³ + 2x.
✏️ Example
Worked: Derivative of x⁴ + x²
Apply the power rule term by term.
↳ Differentiation is linear — handle each term separately, then sum.
💡 Key Idea
Conditions for Maxima and Minima
First derivative zero, second derivative decides.
↳ Stationary point + sign of f'' is the standard 2-mark answer for maxima/minima conditions.
📖 Definition
Partial Derivatives: One Variable at a Time
Freeze the others, differentiate one.
↳ Partial differentiation = ordinary differentiation with other variables held constant.
➗ Formula
Euler's Theorem for Homogeneous Functions
Degree times the function — that's Euler.
Holds only when f is homogeneous of degree n — meaning f(tx, ty) = tⁿ f(x, y).
↳ Euler's theorem is a one-line identity for homogeneous functions of any degree.
📖 Definition
Chain Rule for Multivariable Functions
Outer times inner — every layer.
↳ Chain rule links how f changes to how each input changes.
📖 Definition
Total Derivative vs Partial Derivative
Total derivative counts every path.
↳ Partial derivative freezes variables; total derivative lets them all vary.
💡 Key Idea
Maxima/Minima for Two Variables
Three conditions decide the surface's peaks.
↳ The discriminant D = f_xx·f_yy − (f_xy)² is the two-variable analogue of the second-derivative test.
⚙️ Process
Lagrange's Multiplier Method
Optimize under a constraint in 4 steps.
↳ λ absorbs the constraint so you can optimize f as if it were unconstrained.
➗ Formula
Fundamental Theorem of Calculus
Differentiation and integration undo each other.
↳ Definite integrals are evaluated by finding an antiderivative and subtracting endpoint values.
📖 Definition
Indefinite Integral
Antiderivative plus a constant.
↳ Always add +C — the constant is what makes it 'indefinite'.
📖 Definition
Improper Integrals
Infinite limits or infinite integrands.
↳ Improper integrals are always defined via a limit — never plug in ∞ directly.
➗ Formula
Reduction Formula for ∫ sinⁿx dx
Peel off two powers at a time.
↳ Reduction formulas lower the power by 2 each step until a base case (n = 0 or 1) is reached.
⚠️ Common Mistake
Don't Confuse Partial and Total Derivatives
Freeze vs. follow — know the difference.
↳ The word 'partial' is a promise — only one variable moves.
🎯 Exam Tip
2-Mark Answer Strategy
Definition + formula = full marks.
↳ For 2-mark questions, a crisp definition plus the formula is the safest full-marks pattern.
Smart notes
1
What you'll study, topic by topic
Applied Calculus — Unit 1–3 Important Questions (Part A)
15 minA rapid-fire revision kit covering the core 2-mark concepts from Applied Calculus Units 1–3: derivatives and maxima/minima, partial derivatives and Euler's theorem, and the fundamentals of integral calculus. Each bite ta…
- Applied Calculus uses calculus tools to solve concrete problems in science, engineering, and economics.
- The derivative of $f(x)$ is defined as $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$.
- Geometrically, the derivative is the slope of the tangent line at a point.
Full explanation, worked examples & memory tricks in the app
Leveled MCQ practice
Try the smart MCQs from this kit
All 18 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Warm-up · 8 questions
Q1. Which expression is the limit definition of the derivative of $f(x)$?
Beginner
A
$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$
B
$\lim_{h \to \infty} \frac{f(x+h)-f(x)}{h}$
C
$\frac{f(x+h)-f(x)}{h}$ without any limit
D
$\lim_{h \to 0} \frac{f(x+h)+f(x)}{h}$
Answers and explanations are in the Veda app.
Q2. A curve's tangent line at $x = 3$ has slope 7. What is $f'(3)$?
Beginner
A
Cannot be determined from the tangent
B
7
C
21
D
3
Answers and explanations are in the Veda app.
Q3. Using the power rule, what is $\frac{d}{dx}(x^2)$?
Beginner
A
$x$
B
$x^3/3$
C
$2x^2$
D
$2x$
Answers and explanations are in the Veda app.
Q4. Differentiate $f(x) = x^4 + x^2$.
Beginner
A
$4x^3 + 2x$
B
$x^3 + x$
C
$4x^3 + x$
D
$4x^4 + 2x^2$
Answers and explanations are in the Veda app.
Q5. At an interior extremum of a differentiable function, which condition must hold?
Beginner
A
$f(x) = 0$
B
$f'(x) > 0$
C
$f'(x) = 0$
D
$f''(x) = 0$
Answers and explanations are in the Veda app.
Q6. At a stationary point, $f''(x) < 0$. What does this tell you?
Beginner
A
The point is a local minimum
B
The point is a local maximum
C
The function has no extremum there
D
The test is inconclusive
Answers and explanations are in the Veda app.
Q7. What does the partial derivative $\partial f/\partial x$ measure?
Beginner
A
Change in $f$ as all variables vary together
B
The total change in $f$ along a path in $t$
C
Change in $f$ as $x$ varies, with other variables held constant
D
The value of $f$ when $x = 0$
Answers and explanations are in the Veda app.
Q8. The fundamental theorem of calculus connects differentiation and integration: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$.
Beginner
A
True
B
False
Answers and explanations are in the Veda app.
Building up · 5 questions
Q9. A firm's output $Q(L, K)$ depends on labour and capital. Which derivative tells you the effect of extra labour alone?
Intermediate
A
$dQ/dt$
B
$\partial Q/\partial L$
C
$\partial^2 Q/\partial L \partial K$
D
$\partial Q/\partial K$
Answers and explanations are in the Veda app.
Q10. For $f(x,y)$ with $x = x(t)$ and $y = y(t)$, which formula gives $df/dt$?
Intermediate
A
$f_x \frac{dy}{dt} + f_y \frac{dx}{dt}$
B
$f_x + f_y$
C
$f_x \frac{dx}{dt} - f_y \frac{dy}{dt}$
D
$f_x \frac{dx}{dt} + f_y \frac{dy}{dt}$
Answers and explanations are in the Veda app.
Q11. For a homogeneous function of degree $n$, Euler's theorem states that:
Intermediate
A
$x f_x + y f_y = f$
B
$x f_x + y f_y = n f$
C
$f_x + f_y = n f$
D
$x f_x - y f_y = n f$
Answers and explanations are in the Veda app.
Q12. In the Lagrange multiplier method for optimising $f$ subject to $g = 0$, the Lagrangian is:
Intermediate
A
$\mathcal{L} = f \cdot g$
B
$\mathcal{L} = f + g$
C
$\mathcal{L} = f - \lambda g$
D
$\mathcal{L} = \lambda f - g$
Answers and explanations are in the Veda app.
Q13. What is $\int f(x)\,dx$?
Intermediate
A
A single antiderivative $F(x)$
B
The area under $f$ from $a$ to $b$
C
The derivative of $f(x)$
D
The family of antiderivatives $F(x) + C$
Answers and explanations are in the Veda app.
Topper level · 5 questions
Q14. Which integral is improper?
Advanced
A
$\int_1^{\infty} \frac{1}{x^2}\,dx$
B
$\int_0^{\pi} \sin x\,dx$
C
$\int_1^2 (x+1)\,dx$
D
$\int_0^1 x^2\,dx$
Answers and explanations are in the Veda app.
Q15. An improper integral is defined as a limit of proper integrals. When does it converge?
Advanced
A
When the defining limit exists and is finite
B
When the integrand tends to zero at infinity
C
When the antiderivative is a polynomial
D
When the interval is finite
Answers and explanations are in the Veda app.
Q16. What is the main purpose of the reduction formula for $\int \sin^n x\,dx$?
Advanced
A
Convert the sine integral into a cosine integral permanently
B
Express the integral in terms of a lower power until a base case is reached
C
Evaluate the integral without any further integration
D
Replace the integrand by its Taylor series
Answers and explanations are in the Veda app.
Q17. Given the function $f(x, y) = x^2y + xy^2$, find the total derivative $\frac{df}{dt}$ if $x = t^2$ and $y = 3t$.
Advanced
A
$\frac{df}{dt} = 6t^2 + 6t^3$
B
$\frac{df}{dt} = 2t(3t^2 + 3t) = 6t^3 + 6t^2$
C
$\frac{df}{dt} = 3t(2t + 3)$
D
$\frac{df}{dt} = 3t^2 + 2t^3$
Answers and explanations are in the Veda app.
Q18. Using the Lagrange multiplier method, determine the maximum value of $f(x, y) = 3x + 4y$ subject to the constraint $g(x, y) = x^2 + y^2 - 25 = 0$. What are the coordinates of the maximum point?
Advanced
A
Maximum at $(0, 5)$ with value 20
B
Maximum at $(5, 0)$ with value 15
C
Maximum at $(3, 4)$ with value 24
D
Maximum at $(4, 3)$ with value 25
Answers and explanations are in the Veda app.
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