Differentiation

1. Let f(x) = (function given)
2. To every x, add h: f(x + h)
3. Subtract: f(x + h) − f(x)
4. Divide across by h
5. Let h → 0 → anything with h goes to 0

Tip: Follow the LHS down exactly — the work is in the algebra/trig on the RHS.

dydx = lim
h→0 f(x + h) − f(x)h

This is the same as the change in y over the change in x = slope of the tangent = tan θ.

If y = xⁿ   then   dydx = n · xⁿ⁻¹

★ Multiply by the power, REDUCE the power by one.

Say it over and over — you'll mix it up with integration!

Don't forget: rules of indices, use the tables.

Before you can use the basic rule, get the function into a sum of powers of x:

y = x^a + x^b + x^c + …

• Multiply out any brackets.
• Negative powers: 1a^p = a^−p
• Roots: √x = x^1/2
• Fractions: factorise or split the top over the bottom.

Read the link word between the two parts:

• By (multiplied) ⇒ Product
• Divide by ⇒ Quotient
• Of (a function of a function) ⇒ Chain

(uv)′ = u′v + uv′

★ Leave, Diff + Diff, Leave.

uv ′ = u′v − uv′v² In Tables

Tip: try algebra / factor cancel first! Order matters — v · u′ first, then minus u · v′, then divide by v².

dydx = dudx × dydu

Let u be the inside. Then y becomes a function of u — differentiate both pieces and multiply.

Quick: differentiate INSIDE × differentiate OUTSIDE.

y = sin x ⇒ dydx = cos x

y = cos x ⇒ dydx = −sin x

y = ln x ⇒ dydx = 1x

y = e^x ⇒ dydx = e^x In Tables

ddx ( e^f(x) ) = f′(x) · e^f(x)

→ Diff the power × leave e the same.

ddx ( ln f(x) ) = f′(x)f(x)

→ Diff × (1 over the function).   "Differ the bracket × differ the log."

Break the log up before differentiating:

ln(AB) = ln A + ln B

ln( AB ) = ln A − ln B

ln(Aⁿ) = n ln A

ln e = 1

sin3x = (sin x)³ → chain

ddx ( y² ) = 2y · dydx

y = sin⁻¹( xa ) ⇒ dydx = 1√a² − x²

y = tan⁻¹( xa ) ⇒ dydx = aa² + x² In Tables

a is a constant. Identify it, then drop straight into the format.

y = sin⁻¹( g(x) ) ⇒ dydx = g′(x)√1 − g(x)²

y = tan⁻¹( g(x) ) ⇒ dydx = g′(x)1 + g(x)²

Same pattern as logs: differ the inside on top.

A tangent line needs: a point + a slope.

m = dydx at the point (x, y).

Differentiate y, then sub the x-value of the point in.

y − y₁ = m(x − x₁) In Tables

You need the slope (from dydx at the given x) and a point (the (x, y) where it touches).

• m = y₂ − y₁x₂ − x₁
• m = riserun
• Line ax + by + c = 0 → m = − ab
• m = tan A (angle with positive x-axis, measured anticlockwise)

★ At the point of contact: slope of curve = slope of tangent.

Tangent parallel to a given line → the line's slope = the curve's slope.

Got x? Sub back into the original to get y.

Increasing: dydx > 0

Decreasing: dydx < 0

Stationary / turning: dydx = 0

Max: dydx = 0 AND d²ydx² < 0

Min: dydx = 0 AND d²ydx² > 0

Inflection: d²ydx² = 0

At an inflection the curve isn't bending — it's switching from concave one way to concave the other.

1. What's it the max / min of?
2. Formula: A = …
3. Equation linking the variables
4. Get one variable in terms of the other
5. Sub into step 2, diff, set = 0

Locate the turning points relative to the x-axis:

• Max above the axis, min below → 3 distinct real roots.
• One turning point sits exactly on the axis → 3 real roots, two of them equal.
• Both turning points on the same side of the axis → 1 real root.

• s = f(t) = distance
• v = dsdt = speed
• a = d²sdt² = acceleration
• Initial / at rest: t = 0
• Max speed: dsdt = 0

dVdt = dVdr × drdt

• drdt → m/s
• dAdt → m²/s
• dVdt → m³/s

★ What do we have? What do we need? The LINK is in the question.

Area / volume / surface-area formulas (circle, sphere, cylinder, cube) come straight from the tables.

• f⁻¹ swaps x and y — give the y, find the x.
• f⁻¹ is the image of f under axial symmetry in the line y = x.
• Only a bijective function has an inverse.

1. Write y = f(x).
2. Make x the subject.
3. Write f⁻¹(x).

Quadratics: restrict the domain, complete the square, then solve.