IB Diploma Mathematics: Analysis & Approaches HL · Further Calculus & Differential Equations

Mini-Lesson

Further Calculus & Differential Equations

This HL mini-lesson covers further calculus: implicit differentiation, related rates, integration by substitution and parts, differential equations, Euler's method and Maclaurin series.

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Implicit

Implicit differentiation

When y is defined implicitly, differentiate both sides with respect to x, treating y as a function of x (so d/dx(y²) = 2y·dy/dx).

Worked example

x² + y² = 25 ⇒ 2x + 2y·dy/dx = 0 ⇒ dy/dx = −x/y. At (3, 4): −3/4.

Quick check

Implicit

?Differentiating y² with respect to x gives:

Related rates

Related rates of change

Link the rates of connected quantities with the chain rule. For a sphere, V = (4/3)πr³ gives dV/dr = 4πr², and dV/dt = 4πr²·dr/dt.

Method: write the relationship, differentiate with respect to time, then substitute the known rate and value.

Calculate

Implicit

✎For x² + y² = 25, find dy/dx at the point (3, 4).

Hint: 2x + 2y·dy/dx = 0 ⇒ dy/dx = −x/y = −3/4.

Substitution

Integration by substitution

Reverse the chain rule by letting u = g(x):

∫ f(g(x)) g′(x) dx = ∫ f(u) du

Quick check

By parts

?The integration-by-parts formula is:

Calculate

By parts

✎Evaluate ∫₀¹ x eˣ dx using integration by parts.

Hint: [x eˣ − eˣ]₀¹ = (e − e) − (0 − 1).

By parts

Integration by parts

Reverse the product rule:

∫ u dv = uv − ∫ v du∫₀¹ x eˣ dx = [x eˣ − eˣ]₀¹ = 1

Choosing u: pick u so that du is simpler — logs and powers of x are usually good choices for u.

Sort it

Which integration technique?

Tap an integral, then the best method.

🔁 Substitution

🧩 By parts

➗ Partial fractions

Partial fractions

Integration with partial fractions

Split a rational integrand into partial fractions, then integrate each simple piece (usually to logarithms):

∫ 1/((x)(x+1)) dx = ∫ (1/x − 1/(x+1)) dx = ln|x| − ln|x+1| + C

Calculate

Maclaurin

✎In the Maclaurin series for eˣ, what is the coefficient of x²?

Hint: eˣ = 1 + x + x²/2! + …, so the x² coefficient is 1/2.

Quick check

Separable DE

?To solve a separable differential equation you:

Differential equations

Separable differential equations

Separate the variables, then integrate both sides:

dy/dx = g(x)h(y) ⇒ ∫ dy/h(y) = ∫ g(x) dx

Worked example

dy/dx = 3x², y(0) = 2 ⇒ y = x³ + C ⇒ C = 2, so y(1) = 1 + 2 = 3.

Calculate

Differential equation

✎Solve dy/dx = 3x² with y(0) = 2, then find y(1).

Hint: y = x³ + C, C = 2, so y(1) = 1 + 2.

Match it

Match each rule to its result

Tap a statement on the left, then its matching partner on the right.

Rule

Result

Series methods

Euler’s method & Maclaurin series

Euler's method steps a solution numerically: yₙ₊₁ = yₙ + h·f(xₙ, yₙ). A Maclaurin series expands about x = 0:

eˣ = 1 + x + x²/2! + x³/3! + …coefficient of x² is 1/2

Calculate

Related rate

✎A sphere has V = (4/3)πr³. Find dV/dr at r = 2 to 2 d.p.

Hint: dV/dr = 4πr² = 4π(4) = 16π.

Quick check

Maclaurin

?A Maclaurin series is a Taylor expansion about:

Strategy

Choosing a method

Match structure to tool: f(g(x))g′(x) → substitution; product of unlike functions → by parts; rational integrand → partial fractions; dy/dx factorising into x-part × y-part → separate variables.

Exam habit: for definite integrals by substitution, change the limits to the new variable rather than converting back.

Recap

The big ideas to know

Implicit: d/dx(y²) = 2y dy/dx; solve for dy/dx

Related rates: differentiate the relation w.r.t. time

Integration: substitution / by parts / partial fractions

Differential equations: separate variables and integrate

Series: Euler steps numerically; Maclaurin expands about 0

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