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Differentiation Techniques

Learn differentiation properly: core rules, chain rule patterns, implicit and logarithmic differentiation, plus worked physics-friendly examples.

  • University Physics Year 1
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Differentiation tells you how a quantity changes. In physics, that usually means:

  • Velocity is the derivative of position: v = dx/dt
  • Acceleration is the derivative of velocity: a = dv/dt
  • Field gradients tell you “how fast something changes in space”
Why physics needs this (examples you can click)

1) What a derivative means

The derivative is the slope of the function at a point:

f'(x) = df/dx = lim _(h → 0)(f(x + h)-f(x))/h.

Interpretation: small changes satisfy

Δ f ≈ f'(x) Δ x.
Units check

If x has units of meters and f has units of joules, then df/dx has units of joules per meter.

2) Core rules (the ones you must actually know)

Linearity

(d/dx)(af + bg) = adf/dx + bdg/dx.

Product rule

(d/dx)(uv) = u'v + uv'.

Chain rule (most common source of mistakes)

If y = f(u) and u = u(x), then:

dy/dx = (dy/du)du/dx.

Example pattern:

d/dx sin(3x²) = cos(3x²) · (6x).
Pitfall: the chain-rule factor is not optional

If you differentiate sin(3x²) and write cos(3x²) (forgetting (d/dx)(3x²) = 6x), you are off by a factor that can break units and physics predictions.

Quotient rule

(d/dx)(u/v) = (u'v-uv')/v².

3) Method templates (what to try first)

Derivative workflow (fast + reliable)
  1. Simplify first: rewrite roots as powers, cancel factors, factor constants.
  2. Identify the outermost structure: sum, product, quotient, or composite.
  3. Do the derivative using the correct rule(s): product/quotient plus chain rule.
  4. Simplify the result (factor common terms, combine like terms).
  5. Sanity-check: units, sign, and a quick limit (e.g. x → 0 or t → 0).

Template A: composite functions (chain rule)

If y = f(g(x)), then

dy/dx = f'(g(x))g'(x).

Template B: implicit relationships (constraints)

If F(x,y) = 0, then differentiate both sides:

(d/dx)F(x,y(x)) = 0

and solve for dy/dx.

Template C: “products/powers everywhere” (log differentiation)

If y is a complicated product/quotient/power (especially with variable exponents), take logs:

ln y = ln(expression)

differentiate, then multiply by y to get y'.

4) Techniques beyond the rules

A. Implicit differentiation

If x and y are related by an equation like F(x,y) = 0, you can differentiate both sides with respect to x and treat y as y(x).

B. Logarithmic differentiation

If y is a product/quotient/power that’s hard to differentiate directly, take ln first, differentiate, then solve for y'.

5) Worked examples

Example 1: chain rule

Differentiate y = sin(3x²):

dy/dx = cos(3x²) · 6x = 6x cos(3x²).

Example 2: implicit differentiation (circle)

Given x² + y² = a², find dy/dx.

Differentiate:

2x + 2ydy/dx = 0 ⇒ dy/dx = -x/y.

Example 3: logarithmic differentiation (x^x)

Let y = x^x (for x > 0). Take logs:

ln y = x ln x.

Differentiate:

(1/y)dy/dx = ln x + 1 ⇒ dy/dx = x^x(ln x + 1).

Example 4: a physics-friendly derivative chain

Let x(t) = A cos(ω t). Then:

v(t) = dx/dt = -Aω sin(ω t),

and

a(t) = dv/dt = -Aω² cos(ω t) = -ω² x(t).

This is the defining relationship for simple harmonic motion: a = -ω² x.

Example 5 (physics): force from potential energy

In 1D, if a conservative force comes from a potential U(x), then

Fₓ = -dU/dx.

For gravity (radial coordinate r), a common model is

U(r) = -GMm/r.

Differentiate:

dU/dr = +GMm/r² ⇒ Fᵣ = -dU/dr = -GMm/r².

The minus sign tells you the force points toward decreasing r (attractive).

Example 6 (physics): electric field from a 1D potential

If a potential varies with position as V(x) = V₀e^(-x/a) (with a > 0), then in 1D:

Eₓ = -dV/dx = -(-(V₀/a)e^(-x/a)) = (V₀/a)e^(-x/a).

Units check: volts per meter.

6) Practice (with solutions)

1) Differentiate y = e^{2x} sin(3x)

Use product rule + chain rule:

y' = 2e^2x sin(3x) + e^2x · 3 cos(3x) = e^2x(2 sin(3x) + 3 cos(3x)).
2) Differentiate implicitly: xy + y^2 = 1

Differentiate:

xdy/dx + y + 2ydy/dx = 0.

Collect dy/dx terms:

(x + 2y)dy/dx = -y ⇒ dy/dx = -y/(x + 2y).
3) Differentiate (physics): U(r) = -GMm/r and give F_r

Hint: Fᵣ = -dU/dr.

Answer:

(d/dr)(-GMm/r) = GMm/r² ⇒ Fᵣ = -GMm/r².
4) Use log differentiation: if PV^γ = constant, find dP/dV

Hint: Take logs: ln P + γ ln V = constant.

Answer: Differentiate with respect to V:

(1/P)dP/dV + γ1/V = 0 ⇒ dP/dV = -γP/V.

  • Table of Derivatives

    Use as a cheat sheet once you know the rules.

    Open lesson

  • Integration Techniques

    The complementary skill: antidifferentiation and methods.

    Open lesson

Physics pages where derivatives are the main tool:

Back to Mathematics for Undergraduate Physics