Structured Derivation Template

Example 1—Expanding a quadratic expression

(x+y)^2

Step 1—Write as a product

(x+y)*(x+y)

Step 2—Expand

x*(x+y)+y*(x+y)

Step 3—Distribute

x^2+x*y+x*y+y^2

Step 4—Combine terms

x^2+2*x*y+y^2

Result

(x+y)^2=x^2+2*x*y+y^2

Example 2—Derivative

Problem

Compute d(x^3)/d(x)

Step 1—Use derivative definition

(lim_h→0)(((x+h)^3-x^3)/h)

Step 2—Expand the cube

(x+h)^3=x^3+3*x^2*h+3*x*h^2+h^3

Step 3—Substitute

(x^3+3*x^2*h+3*x*h^2+h^3-x^3)/h

Step 4—Simplify

3*x^2+3*x*h+h^2

Step 5—Take the limit

3*x^2

Result

d(x^3)/x=3*x^2

Example 3—Matrix multiplication

Matricies

A=[[1,2],[3,4]] B=[[2,1],[0,3]]

Multiply

A*B=[[1⋅2+2⋅0,1⋅1+2⋅3],[3⋅2+4⋅0,3⋅1+4⋅3]]

Result

A*B=[[2,7],[6,15]]

Example 4—Differential equation

Equation

d(P)/d(t)=k*P

Step 1—Separate variables

d(P)/P=k*d(t)

Step 2—Integrate

(∫_^)(d(P)/P=(∫_^)(k*d(t)))

Step 3—Evaluate

ln(P)=k*t+C

Step 4 — Solve for P(t)

P(t)=C*ℇ^(k*t)

Result

If P(0)=(P_0), P(t)=(P_0)*ℇ^(k*t)