Calculus Section 2.2 Basic Differentiation Rules and Rates of Change.
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Transcript of Calculus Section 2.2 Basic Differentiation Rules and Rates of Change.
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)