Application of Derivatives — Part 1: Rates of change; increasing and decreasing functions: visual lesson, CBSE Class 12 Mathematics

Lesson

Application of Derivatives, Part 1

How fast things change, and which way they move.

In this lesson you will

Interpret a derivative as a rate of change
Solve related-rate problems with the chain rule
Find marginal cost and marginal revenue
Find intervals on which a function is strictly increasing or decreasing

$y = x^3 - 3x^2 - 9x + 5$

Rising, falling, rising: the sign of the derivative tells us which.

Concept

What you will be able to do

Read a derivative as a rate of change
Solve related rates problems
Find marginal cost and marginal revenue
Find intervals where $f$ is strictly increasing or strictly decreasing

Concept

The derivative is a rate of change

If $y = f(x)$ , then $\dfrac{dy}{dx}$ is the rate of change of $y$ with respect to $x$ : how much $y$ changes per unit change in $x$ , at that instant.

Every derivative is a rate of change of one quantity with respect to another.
Quantity	Derivative	Meaning
distance $s(t)$	$\dfrac{ds}{dt}$	velocity
area $A(r) = \pi r^2$	$\dfrac{dA}{dr} = 2\pi r$	growth of area per unit of radius
total cost $C(x)$	$C'(x)$	marginal cost
revenue $R(x)$	$R'(x)$	marginal revenue

Cookies