Power Rule for Derivatives
d/dx(xⁿ) = n × x^(n-1)
Find the derivative of x raised to the power n
Definite Integral (Power Rule)
∫[a to b] x^n dx = [x^(n+1)/(n+1)]|[a to b]
Calculate definite integral using power rule
Partial Derivative
∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/h
Calculate partial derivative ∂(x²y)/∂x = 2xy
Chain Rule (Derivative)
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Find the derivative of a composite function. Example: d/dx[(axⁿ)ᵐ] evaluated at x.
Product Rule (Derivative)
d/dx[f·g] = f'·g + f·g'
Differentiate a product of two power functions: f(x) = xᵃ and g(x) = xᵇ at a given x.
Quotient Rule (Derivative)
d/dx[f/g] = (f'·g − f·g') / g²
Differentiate f(x)=xᵃ divided by g(x)=xᵇ at a given x (requires b ≠ a).
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Derivative of sin(ax)
d/dx[sin(ax)] = a·cos(ax)
Compute the derivative of sin(ax) at a specific x value (x in radians).
Derivative of cos(ax)
d/dx[cos(ax)] = −a·sin(ax)
Compute the derivative of cos(ax) at a specific x value (x in radians).
Derivative of eᵃˣ
d/dx[eᵃˣ] = a·eᵃˣ
Compute the derivative of the exponential function eᵃˣ at a specific x.
Derivative of ln(ax)
d/dx[ln(ax)] = 1/x
Compute the derivative of the natural logarithm ln(ax) at a given positive x.
Implicit Differentiation (Circle)
x² + y² = r² → dy/dx = −x/y
Find dy/dx implicitly for the circle equation x² + y² = r² at point (x₀, y₀).
Critical Points (Polynomial)
f'(x) = 0 → Critical Points
Find the critical point of f(x) = ax³ + bx² + cx by solving f'(x) = 0 (returns one root).
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Higher-Order Derivative (xⁿ)
dᵏ/dxᵏ(xⁿ) = n!/(n−k)! · x^(n−k)
Compute the k-th order derivative of xⁿ evaluated at x.
L'Hôpital's Rule
lim f(x)/g(x) = lim f'(x)/g'(x) [0/0 form]
Evaluate lim[x→a] (xⁿ − aⁿ)/(x − a) using L'Hôpital's Rule (= n·aⁿ⁻¹).
Taylor Series (eˣ, n terms)
eˣ ≈ Σₖ₌₀ⁿ xᵏ/k!
Approximate eˣ using the Taylor series expansion centred at 0, with n terms.
Mean Value Theorem
f'(c) = (f(b) − f(a)) / (b − a)
Find the value c guaranteed by MVT for f(x) = xⁿ on [a, b] (returns average rate of change).
Indefinite Integral of sin / cos
∫sin(ax)dx = −cos(ax)/a + C ; ∫cos(ax)dx = sin(ax)/a + C
Evaluate ∫sin(ax)dx or ∫cos(ax)dx. Choose type: 1 = sin, 2 = cos.
Integration by Parts (xⁿ·eˣ)
∫u dv = uv − ∫v du
Compute ∫[a,b] xⁿ·eˣ dx using integration by parts repeatedly.
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U-Substitution
∫f(g(x))g'(x)dx — let u = g(x)
Compute ∫[a,b] (ax+b)ⁿ dx via substitution u = ax+b.
Riemann Sum (Left/Right/Mid)
∫[a,b]f(x)dx ≈ Σᵢ f(xᵢ*)·Δx
Approximate ∫[a,b] xⁿ dx using n rectangles. Type: 1=Left, 2=Right, 3=Midpoint.
Trapezoidal Rule
∫[a,b]f dx ≈ (Δx/2)[f(x₀)+2f(x₁)+…+2f(xₙ₋₁)+f(xₙ)]
Approximate ∫[a,b] xⁿ dx using the Trapezoidal Rule with N subintervals.
Simpson's Rule
∫[a,b]f dx ≈ (Δx/3)[f(x₀)+4f(x₁)+2f(x₂)+…+4f(xₙ₋₁)+f(xₙ)]
Approximate ∫[a,b] xᵖ dx using Simpson's Rule. N must be even.
Arc Length (Polynomial)
L = ∫[a,b] √(1 + [f'(x)]²) dx
Approximate arc length of f(x) = xⁿ on [a,b] via numerical integration (100 subintervals).
Surface of Revolution (about x-axis)
S = 2π ∫[a,b] f(x)√(1+[f'(x)]²) dx
Approximate the surface area when f(x) = xⁿ is revolved around the x-axis over [a,b].
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Improper Integral (1/xᵖ, 1 to ∞)
∫[1,∞] 1/xᵖ dx = 1/(p−1) if p > 1, else diverges
Determine convergence and value of ∫[1,∞] x^(−p) dx.
Area Between Two Curves
A = ∫[a,b] |f(x) − g(x)| dx
Compute the area between f(x) = xᵃ and g(x) = xᵇ on [0, 1] (assumes f ≥ g on interval).
Volume of Revolution (Disk Method)
V = π ∫[a,b] [f(x)]² dx
Calculate the volume when f(x) = xⁿ is revolved about the x-axis over [a, b].
Volume of Revolution (Shell Method)
V = 2π ∫[a,b] x · f(x) dx
Calculate the volume when f(x) = xⁿ is revolved about the y-axis using cylindrical shells.
Gradient Vector ∇f
∇f = (∂f/∂x, ∂f/∂y) for f = axⁿyᵐ
Compute the gradient vector of f(x,y) = a·xⁿ·yᵐ at point (x₀, y₀).
Directional Derivative
Dᵤf = ∇f · û
Compute the directional derivative of f(x,y) = ax²y² in direction (u₁, u₂) at (x₀, y₀).
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Double Integral (Rectangle)
∬_R f(x,y) dA = ∫[c,d] ∫[a,b] f(x,y) dx dy
Compute ∬ f(x,y) = xⁿ·yᵐ over rectangle [a,b]×[c,d].
Jacobian Determinant (2×2)
J = det ∂(u,v)/∂(x,y) = ∂u/∂x · ∂v/∂y − ∂u/∂y · ∂v/∂x
Compute the Jacobian for the transformation u=ax+by, v=cx+dy.
Divergence ∇·F (2D)
∇·F = ∂P/∂x + ∂Q/∂y for F = (P, Q) = (axⁿyᵐ, bxᵖyq)
Compute divergence of F = (x^n, y^m) at point (x₀, y₀).
Curl ∇×F (2D scalar)
∇×F = ∂Q/∂x − ∂P/∂y for F = (P(x,y), Q(x,y))
Compute the 2D scalar curl of F = (P, Q) = (y^n, x^m) at (x₀, y₀).
Separable ODE: dy/dx = ky
dy/dx = ky → y = C·eᵏˣ
Solve separable ODE dy/dx = ky with initial condition y(0) = y₀. Evaluate at x.
First-Order Linear ODE (Integrating Factor)
y' + P(x)y = Q(x) → y = e^(−∫P dx)[∫Q·e^(∫P dx)dx + C]
Solve y' + py = q (constant P=p, Q=q) with y(0) = y₀. Evaluates at x.
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2nd-Order Homogeneous ODE
ay'' + by' + cy = 0 → characteristic equation: ar² + br + c = 0
Solve ay''+by'+cy=0 with y(0)=1, y'(0)=0. Evaluate at x.
Euler's Method (ODE Approximation)
yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
Approximate y(x_target) for dy/dx = ky using Euler's method with step size h.
Exponential Growth / Decay ODE
dP/dt = kP → P(t) = P₀eᵏᵗ
Model population growth (k > 0) or radioactive decay (k < 0). Find P at time t.
Limit of a Polynomial
lim[x→a] f(x) — direct substitution for polynomials
Evaluate lim[x→a] (xⁿ + bx + c) by direct substitution (polynomial).
Squeeze Theorem (sin(x)/x)
lim[x→0] sin(x)/x = 1
Demonstrate the Squeeze Theorem for lim[x→0] sin(x)/x. Evaluates sin(h)/h near h = 0.
Continuity Check
f continuous at a ⟺ lim[x→a] f(x) = f(a)
Check continuity of f(x) = xⁿ at x = a. Verifies all three conditions for continuity.
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