Calculus

Section 2.2

Basic Differentiation Rules and Rates of Change

The Constant Rule

d

dxc 0

For every x value, the slope is always 0. Therefore, the derivative of a constant function is 0.

What is the slope of the graph of the function f(x) = 6?

Example : f (x) 3

f (x) 0

The Power Rule

If n is a rational number, then the function f (x) x n

is differentiable and

d

dxx n nx n 1

Example : f (x) x 4

Multiply the exponent to the

coefficient and

reduce the exponent

by 1.

f (x) 4 x 3

The Constant Multiple Rule

If f is a differentiable function, and c is a real number,

then cf is also differentiable and

d

dxcf (x) c f (x)

Multiply the coefficient

by the derivative.

Example : f (x) 3x 4

f (x) 3(4x 3)

f (x) 12x 3

The Sum and Difference Rules

The sum (or difference) of two differentiable functions

is differentiable and is the sum (or difference)

of their derivatives.

d

dxf (x) g(x) f (x) g (x)

d

dxf (x) g(x) f (x) g (x)

Derivatives of Polynomials

Using the various differentiation rules,

one can now derive a polynomial.

Examples :

f (x) 3x 2

g(x) 5x 4 4x 3 7x 2 3

g (x) 20x 3 12x 2 14x

f (x) 3(2x) 6x

A Constant Times a Variable to an Exponent

Rewrite the function to get the terms as a constant times a variable to an exponent.

A CONSTANT TIMES A VARIABLE TO AN EXPONENT!!!!

Example :

f (x) 3

x 2

f (x) 3x 2

f (x) 6x 3

f (x) 6

x 3

Radicals to Rational Exponents

In order to derive a radical function,

change it into a rational exponent,

then apply the Power Rule.

x nm xnm

Derivatives of Sine and Cosine Functions

d

dxsin x cos x

d

dxcos x sin x

Examples :

f (x) 2sin x

f (x) 2cos x

y 3x 2 cos x

y 6x ( sin x)

Rates of Change

The derivative is the rate of change of one variable with respect to another.

Usually we talk about the rate of change of y with respect to x.

dy

dx

Vertical Motion of an Object

The function s(t) that gives the position of an object (relative to the origin) as a function of time t is called the position function.

Position Function

s(t) 12 gt 2 v0t s0

Where g is the acceleration due to gravity, v0 is the initial velocity, and s0 is the initial position of the object.

Average Velocity

Rate distance

time

Average Velocity s

t

Instantaneous Velocity

The velocity function is the derivative of the position function (i.e. the rate of change of the position function at any instant of time t).

v(t) s (t)

Acceleration

The acceleration function is the derivative of the velocity function (i.e. the rate of change of the velocity function at any time t).

a(t) v (t) s (t)