Calc III Derivatives and Antiderivatives

Power Rule

d/dx (xⁿ) = n·xⁿ⁻¹

Constant Multiple Rule

d/dx (c·g(x)) = c·g'(x)

Derivative of a constant

0

Derivative of f + g

f’ + g’

Derivative of √x

1/(2√x)

Derivative ln(x)

1/x

sin(x) derivative line

sin(x) → cos(x) → -sin(x) → -cos(x) → sin(x)

derivative of tan(x)

sec²(x)

Product Rule

(f·g)’ = f’·g + f·g’

Quotient Rule

(f/g)’ = (f’·g − f·g’) / g²

Chain Rule

d/dx [f(g(x))] = f'(g(x)) · g'(x)

Power Rule (Reverse)

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (for n ≠ −1)

Constant Coefficient (Reverse)

cos(kx) → sin(kx) / k + C

u substitution

Antiderivative of √x

2/3𝑥3/2 + 𝐶⁠

Antiderivative of lnx

xlnx - x + C

Derivative of ln(x/y)

Derivative of ln(x) - ln(y)

F(x, y) = 0 where y is differentiable of x

dy/dx = -Fx/Fy

F(x, y, z) = 0

dz/dx = -Fx/Fz dz/dy = -Fy/Fz

Directional Derivatives

u must be a unit vector (length 1). If the direction vector you’re given isn’t already length 1, divide it by its length first.

u / (magnitude of u) dotproduct (Fx, Fy, Fz) with respect to P.

Local Extrema

1. Find critical points: set f_x = 0 and f_y = 0 and solve the system.

2. Compute the discriminant: D = f_xx · f_yy − (f_xy)²

3. Classify each critical point:

   • D > 0 and f_xx > 0 → local minimum

   • D > 0 and f_xx < 0 → local maximum

   • D < 0 → saddle point (neither)

   • D = 0 → test is inconclusive

Direction of vector v

u = v/|v|

Tangent Line to a level curve

fx(x₀,y₀)(x-x₀) + fy(x₀,y₀)(y-y₀) = 0

Where the function increases most rapidly/Where it decreases most rapidly (each vector divided by its length

(Fxi + Fyj)|_{P(x,y) , -}(Fxi + Fyj)|_P(x,y) (The magnitude, not the vector!)

Tangent plane to f(x, y, z) = c at P₀

fx(P₀)(x-x₀) + fy(P₀)(y-y₀) + fz(P₀)(z-z₀) = 0

Normal line to f(x, y, z) = c at P₀

x = x₀ +fx(P₀)t, y = y₀ +fy(P₀)t, z = z₀ +fz(P₀)t

Plane tangent to Surface → z = f(x, y)

fx(x₀,y₀)(x-x₀) + fy(x₀,y₀)(y-y₀) - (z-z₀) = 0

Estimating the change of f

Single variable → f’(P₀)ds Two or more variables → (Directional Derivative dotprod u)ds (u = (P₀P₁→) / |P₀P₁|)

The linearization of a function f(x, y)

L(x, y) = f(x₀, y₀) + fx(x₀,y₀)(x-x₀) + fy(x₀,y₀)(y-y₀)

Standard Linear Approximation of f at (x₀, y₀)

f(x, y) ≈ L(x, y)