Derivative Notation and Meaning

Learn derivative units and notation, second derivatives.

Lesson 3 of 32 parts Practice Flashcards

1/2

Derivative Units and Notation

Objective: Explain derivative units and notation, apply the stated method, and check its conditions.

The derivative dQ/dt has units of Q divided by units of t. It is distinct from Q itself. Prime notation and Leibniz notation describe the same derivative when variables are clear; a derivative value requires a specified input. A quantity and its rate use different units.

Formulas

Derivative units = output units/input units

Worked examples

Try itA quantity Q(t)=6t²+7 is measured in liters with t in minutes. Find Q′(7) in liters/minute.Show answer

Step 1: Differentiate Q′(t)=12t. Step 2: Substitute 7 minutes. Step 3: The rate is 84 liters/minute.

Swipe sideways to move between sections

Derivative Units and Notation

Objective: Explain derivative units and notation, apply the stated method, and check its conditions.

Formulas

Derivative units = output units/input units

Worked examples

Try itA quantity Q(t)=6t²+7 is measured in liters with t in minutes. Find Q′(7) in liters/minute.Show answer

Step 1: Differentiate Q′(t)=12t. Step 2: Substitute 7 minutes. Step 3: The rate is 84 liters/minute.

Second Derivatives

Objective: Explain second derivatives, apply the stated method, and check its conditions.

The second derivative f″ is the derivative of f′. It measures change in the first derivative and has units output per input squared. For position in time, s″ is acceleration; zero acceleration means constant velocity, not necessarily rest. Zero acceleration does not require zero velocity.

Formulas

$f″=(f′)′$

Worked examples

Try itFor f(x)=6x³+7x, find f″(7).Show answer

Step 1: First derivative: f′(x)=18x²+7. Step 2: Second derivative: f″(x)=36x. Step 3: f″(7)=252.

You finished the reading!

Lock it in with a few practice questions. This is the last lesson in the chapter.

Practice

Previous lessonTangent Lines

Last lesson