Calculus has a reputation: hard, abstract, only for math geniuses.
The truth: Calculus is just fancy addition and subtraction. Once you understand the intuition, the formulas make sense.
This guide strips away the complexity and reveals what calculus is really about.
Calculus answers two questions:
1. Derivatives: How fast is something changing?
What is a derivative? The slope of a curve at a specific point.
Remember slope from algebra: slope = rise/run = (y₂ - y₁)/(x₂ - x₁)
For a straight line: Slope is constant.
For a curve: Slope changes at every point. The derivative tells you the slope at any specific point.
Position vs. Time:
At any moment, the derivative tells you how fast position is changing.
For f(x) = xⁿ: f'(x) = nxⁿ⁻¹
Examples:
1. f(x) = x² f'(x) = 2x²⁻¹ = 2x¹ = 2x
2. f(x) = x³ f'(x) = 3x³⁻¹ = 3x²
3. f(x) = x⁴ f'(x) = 4x³
Pattern:
f(x) = x f'(x) = 1x⁰ = 1
f(x) = 5 (constant) f'(x) = 0 (slope of horizontal line is zero)
Constant Multiple Rule: If f(x) = c · g(x), then f'(x) = c · g'(x)
Example: f(x) = 3x² f'(x) = 3 · (2x) = 6x
Sum Rule: If f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x)
Example: f(x) = x³ + 2x² f'(x) = 3x² + 4x
Product Rule: If f(x) = g(x) · h(x), then f'(x) = g'(x)·h(x) + g(x)·h'(x)
Chain Rule (for compositions): If f(x) = g(h(x)), then f'(x) = g'(h(x)) · h'(x)
GPAI tip: Confused about which rule to use? Upload your derivative problem to GPAI.
Example 1: Find critical points (maxima/minima)
Problem: f(x) = -x² + 4x Find the maximum value.
Solution: Step 1: Take derivative f'(x) = -2x + 4
Step 2: Set derivative = 0 -2x + 4 = 0 x = 2
Step 3: Check if it's a max or min f''(x) = -2 (negative, so it's a maximum)
Step 4: Find maximum value f(2) = -(2)² + 4(2) = -4 + 8 = 4
Answer: Maximum value is 4 at x = 2
Example 2: Related rates
Problem: A balloon is being inflated. Its radius is increasing at 2 cm/s. How fast is the volume increasing when radius = 5 cm?
Given:
Take derivative with respect to time: dV/dt = (4/3)π · 3r² · dr/dt dV/dt = 4πr² · dr/dt
Substitute: dV/dt = 4π(5)² · 2 dV/dt = 4π(25) · 2 dV/dt = 200π cm³/s
Answer: Volume is increasing at 200π cm³/s
What is an integral? The total area under a curve between two points.
Why it's useful:
Derivative: f(x) = x² → f'(x) = 2x Integral: ∫2x dx = x² + C
The "+C": Since derivatives of constants = 0, we add "C" (could be any constant).
For ∫xⁿ dx: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (where n ≠ -1)
Examples:
1. ∫x² dx = x³/3 + C
2. ∫x³ dx = x⁴/4 + C
3. ∫5x⁴ dx = 5 · x⁵/5 + C = x⁵ + C
Pattern:
Notation: ∫ᵃᵇ f(x) dx
Means: Area under f(x) from x=a to x=b
How to solve: 1. Find antiderivative F(x) 2. Evaluate F(b) - F(a)
Example: ∫₀² x² dx
Step 1: Antiderivative of x² is x³/3
Step 2: Evaluate at bounds = [x³/3] from 0 to 2 = (2³/3) - (0³/3) = 8/3 - 0 = 8/3
Answer: 8/3
Example 1: Area between curves
Problem: Find area between y = x² and y = x from x=0 to x=1
Solution: Area = ∫₀¹ (top function - bottom function) dx = ∫₀¹ (x - x²) dx = [x²/2 - x³/3] from 0 to 1 = (1/2 - 1/3) - (0 - 0) = 3/6 - 2/6 = 1/6
Answer: 1/6 square units
Example 2: Total distance
Problem: A car's velocity is v(t) = 3t² (m/s). How far does it travel from t=0 to t=4 seconds?
Solution: Distance = ∫₀⁴ v(t) dt = ∫₀⁴ 3t² dt = [t³] from 0 to 4 = 4³ - 0³ = 64 meters
Answer: 64 meters
Connects derivatives and integrals:
If F(x) is an antiderivative of f(x), then: ∫ᵃᵇ f(x) dx = F(b) - F(a)
What this means:
Process: 1. Write function to optimize 2. Take derivative 3. Set derivative = 0 4. Solve for critical points 5. Test which is max/min
When one quantity affects another:
Position → Velocity → Acceleration
Integrals calculate:
Before memorizing formulas, understand:
Master these first:
For derivatives: Sketch the curve, visualize the tangent line (slope = derivative)
For integrals: Sketch the curve, shade the area under it
Derivative check: Take derivative of your answer. Should get back original function.
Integral check: Take integral of your derivative. Should get back original function (plus C).
Upload calculus problems:
❌ Forgetting +C in indefinite integrals ∫x² dx = x³/3 + C (NOT just x³/3)
❌ Wrong exponent arithmetic Power rule: Multiply by exponent, then subtract 1 (NOT the other way around)
❌ Ignoring bounds in definite integrals Must evaluate at upper bound minus lower bound
❌ Mixing up derivative and integral rules Derivative: Exponent comes down Integral: Exponent goes up
Calculus isn't abstract magic. It's answering two practical questions: 1. How fast is it changing? (Derivative) 2. How much has accumulated? (Integral)
Master the basics:
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