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Calculus and Change

Subject: Mathematics
Topic: 4
Cambridge Code: 0580

Differentiation​

Differentiation - Finding rate of change (derivative)

Concept​

Derivative f′(x)f'(x) - Rate of change of f(x)

  • Instantaneous rate of change
  • Slope of tangent line
  • Velocity (if x is time)

Definition​

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Differentiation Rules​

Power rule: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}

Example: ddx(x3)=3x2\frac{d}{dx}(x^3) = 3x^2

Constant rule: ddx(c)=0\frac{d}{dx}(c) = 0

Constant multiple: ddx(cf(x))=c⋅f′(x)\frac{d}{dx}(cf(x)) = c \cdot f'(x)

Sum rule: ddx(f(x)+g(x))=f′(x)+g′(x)\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)

Product rule: ddx(f(x)g(x))=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)

Quotient rule: ddx(f(x)g(x))=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain rule: ddx(f(g(x)))=f′(g(x))⋅g′(x)\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)

Derivatives of Special Functions​

ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x ddx(ex)=ex\frac{d}{dx}(e^x) = e^x ddx(ln⁡x)=1x\frac{d}{dx}(\ln x) = \frac{1}{x}

Applications of Differentiation​

Finding Stationary Points​

Stationary points - Where f′(x)=0f'(x) = 0

Types:

  • Maximum: f′′(x)<0f''(x) < 0
  • Minimum: f′′(x)>0f''(x) > 0
  • Inflection: f′′(x)=0f''(x) = 0 (changes concavity)

Second Derivative Test​

f′′(x)=ddx(f′(x))f''(x) = \frac{d}{dx}(f'(x))

  • f′′(x)>0f''(x) > 0: Concave up (minimum)
  • f′′(x)<0f''(x) < 0: Concave down (maximum)
  • f′′(x)=0f''(x) = 0: Possible inflection point

Optimization​

To find maximum/minimum value:

  1. Find f′(x)f'(x)
  2. Set f′(x)=0f'(x) = 0, solve for x
  3. Find corresponding y-value
  4. Verify with second derivative or context

When multiple variables change together:

dVdt=dVdr⋅drdt\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}

Motion Problems​

Position: s(t)s(t) (distance at time t) Velocity: v(t)=dsdtv(t) = \frac{ds}{dt} (rate of position change) Acceleration: a(t)=dvdt=d2sdt2a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} (rate of velocity change)

Integration​

Integration - Reverse of differentiation; finding area

Basic Integration​

Indefinite integral: ∫xn dx=xn+1n+1+C,n≠−1\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n ≠ -1

Constant of integration: C (arbitrary constant)

Integration Rules​

Sum rule: ∫(f(x)+g(x)) dx=∫f(x) dx+∫g(x) dx\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx

Constant multiple: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx = c \int f(x) \, dx

Integrals of Special Functions​

∫ex dx=ex+C\int e^x \, dx = e^x + C ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C ∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C ∫cos⁡x dx=sin⁡x+C\int \cos x \, dx = \sin x + C

Definite Integration​

Definite integral - Area between curve and x-axis

Fundamental Theorem of Calculus​

∫abf(x) dx=[F(x)]ab=F(b)−F(a)\int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a)

Where F is antiderivative of f

Area Under Curves​

Between curve and x-axis: Area=∫abf(x) dx\text{Area} = \int_a^b f(x) \, dx

Between two curves: Area=∫ab∣f(x)−g(x)∣ dx\text{Area} = \int_a^b |f(x) - g(x)| \, dx

Numerical Integration​

Trapezoid rule: ∫abf(x) dx≈h2[f(x0)+2f(x1)+2f(x2)+...+2f(xn−1)+f(xn)]\int_a^b f(x) \, dx ≈ \frac{h}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]

Where h=b−anh = \frac{b-a}{n}

Differential Equations​

Differential equation - Equation involving derivatives

Separable Equations​

dydx=f(x)g(y)\frac{dy}{dx} = f(x)g(y)

Solution: ∫dyg(y)=∫f(x) dx\int \frac{dy}{g(y)} = \int f(x) \, dx

Example​

dydx=2x,y(0)=1\frac{dy}{dx} = 2x, \quad y(0) = 1 y=x2+1y = x^2 + 1

Key Points​

  1. Derivative is rate of change
  2. Power rule most common differentiation rule
  3. Chain rule for composite functions
  4. Stationary points where f'(x) = 0
  5. Second derivative determines max/min
  6. Integration is reverse of differentiation
  7. Definite integral gives area
  8. Trapezoid rule approximates area
  9. Related rates connect changing quantities
  10. Differential equations model real processes

Practice Questions​

  1. Differentiate using various rules
  2. Find stationary points
  3. Determine max/min values
  4. Solve optimization problems
  5. Integrate various functions
  6. Evaluate definite integrals
  7. Find areas under curves
  8. Find areas between curves
  9. Solve motion problems
  10. Solve differential equations

Revision Tips​

  • Master differentiation rules thoroughly
  • Practice chain rule extensively
  • Understand what derivative represents
  • Know special function derivatives
  • Practice optimization problems
  • Understand integration as area
  • Learn definite integral evaluation
  • Connect to real-world applications
  • Verify answers by working backward