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§ Calculus

Differentiation

§ Calculus

Differentiation

CCSS.HSF.IF3 min read

Differentiation is the mathematical process of finding the derivative of a function, which represents the instantaneous rate of change at any given point. The derivative of f(x) = 3x² is f'(x) = 6x, meaning at x = 2, the function changes at a rate of 12 units per unit input. This fundamental operation in calculus transforms polynomials, trigonometric functions, and exponentials into expressions that describe their slopes.

§ 01

Why it matters

Differentiation appears throughout physics, engineering, and economics to model changing quantities. In physics, differentiating position functions gives velocity — if an object's position is s(t) = 16t², its velocity is s'(t) = 32t feet per second. Engineers use derivatives to optimize bridge designs and minimize material costs. In economics, marginal cost functions derive from total cost functions through differentiation — if producing x items costs C(x) = 0.01x³ + 2x + 100 dollars, the marginal cost is C'(x) = 0.03x² + 2 dollars per additional item. Calculus courses build on differentiation to explore integration, differential equations, and multivariable calculus, making it essential preparation for advanced STEM fields.

§ 02

How to solve differentiation

Differentiation

  • Power rule: d/dx [xⁿ] = nxⁿ⁻¹.
  • Chain rule: d/dx [f(g(x))] = f'(g(x)) × g'(x).
  • Product rule: d/dx [uv] = u'v + uv'.
  • Derivative = gradient of the tangent = instantaneous rate of change.

Example: f(x) = 3x⁴ → f'(x) = 12x³. At x=2: f'(2) = 96.

§ 03

Worked examples

Beginner§ 01

Differentiate: f(x) = 3 x2

Answer: f'(x) = 6 x

  1. Apply the power rule: d/dx[axn] = nax(n-1) → f'(x) = 2·3x1 = 6 x — Multiply the exponent 2 by the coefficient 3, then reduce the exponent by 1.
Easy§ 02

Differentiate: f(x) = 3 x3 + 4 x2 - 3 x - 3

Answer: f'(x) = 9 x2 + 8 x - 3

  1. Write out the rule → d/dx[xn] = n·x(n-1) — The power rule: multiply by the exponent, then reduce the exponent by 1.
  2. Differentiate 3 x3 → 3·3x2 = 9 x2 — Exponent 3 comes down as a factor, exponent becomes 3−1 = 2.
  3. Differentiate 4 x2 → 2·4x = 8 x — Exponent 2 comes down, exponent becomes 2−1 = 1.
  4. Differentiate -3x → -3 — The derivative of kx is just k. The constant d vanishes.
  5. Combine all terms → f'(x) = 9 x2 + 8 x - 3 — Write the derivative as one expression.
Medium§ 03

Differentiate: f(x) = 2 sin(x)

Answer: f'(x) = 2 cos(x)

  1. Apply the rule: d/dx[sin(x)] = cos(x) → f'(x) = 2 cos(x) — The constant 2 is carried through.
§ 04

Common mistakes

  • A common error with the power rule is writing the derivative of x³ as 3x³ instead of 3x², forgetting to reduce the exponent by 1 after bringing it down as a coefficient.
  • When differentiating 2x⁴ + 5x, a frequent mistake is writing 8x⁴ + 5 instead of 8x³ + 5, applying the power rule incorrectly to the linear term.
  • With the chain rule, differentiating (3x + 1)² often produces 2(3x + 1) instead of 6(3x + 1), missing the derivative of the inner function 3x + 1.
§ 05

Frequently asked questions

What is the difference between a derivative and differentiation?
Differentiation is the process or operation of finding a derivative, while a derivative is the result — the function that represents the rate of change. For f(x) = x³, differentiation produces the derivative f'(x) = 3x².
How do you check if a derivative is correct?
Verify derivatives by checking specific points or using differentiation rules backwards. For f(x) = 2x³ with derivative f'(x) = 6x², substitute x = 2: f'(2) = 24, which should match the slope of the tangent line at that point.
Why does the derivative of a constant equal zero?
Constants don't change, so their rate of change is zero. The graph of f(x) = 7 is a horizontal line with slope 0 everywhere, making f'(x) = 0. This follows from the power rule: 7 = 7x⁰, so the derivative is 0 × 7x⁻¹ = 0.
What does it mean when a derivative is negative?
A negative derivative indicates the function is decreasing at that point. For f(x) = -x² + 4, the derivative f'(x) = -2x is negative when x > 0, showing the parabola slopes downward on the right side of the vertex.
When do you use the chain rule versus the power rule?
Use the power rule for simple powers like x⁵ or 3x⁴. Use the chain rule when the base contains a function, like (2x + 3)⁴ or sin(3x). The chain rule handles composite functions where one function is inside another.
§ 06

See also

FunctionExponentCoefficientTermSlopeMeanVertexBase
§ 06

Where to next?

Prerequisites
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