by: jenn gulya the derivative of a function f with respect to the variable is the function f ‘...
TRANSCRIPT
By: Jenn Gulya
dx
dy
The derivative of a function f with respect to the variable is the function f ‘ whose value at x, if the limit exists, is:
dx
dy
h
xfhxfh
)()(lim
0
This value is, also, representative of the slope of the function at a point.
Where a is an x value, F(a) must be differentiable for f’(a) to exist. So at some points a derivative may not exist.
(An example would be the absolute value graph, which lacks a derivative at x=0).
1. Derivative of a Constant Function:
If f is the function with a constant value c, then f’ = 0
2. Power Rule for Positive/ Negative Integer Powers of x:
If n is a positive/negative integer, then )1()(
dx
d nn nxx
3. The Constant Multiple Rule If u is a differentiable function of x and c is a constant, then
4. The Sum and Difference Rule If u and v are differentiable function of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such point
dx
duccu )(
dx
d
dx
dv
dx
duvu )(
dx
d
5.The Product Rule5.The Product RuleThe product of two differentiable functions The product of two differentiable functions uu and and vv is differentiable, and is differentiable, and
6. The Quotient RuleAt a point where v 0, the quotient y= u/v of two differentiable functions is differentiable, and
dx
du v
dx
dvu (uv)
dx
d
2vdxdv
udxdu
v
v
u
dx
d
Y= x3+6x2-(5/3)x+16
Applying Rules 1 through 4, differentiate the polynomial term-by-term
)3/5(123
0)3/5(2*63
)16()3/5()6()(
2
2
23
xx
xx
dx
dx
dx
dx
dx
dx
dx
d
dx
dy
Graph of original Function:
Graph of derivative: It can be seen that when the original curve levels off and changes direction the graph of f’ crosses the x-axis.
Calculations can be graphically supported.
Differentiate f(x)=1
12
2
x
x
22
22
33
22
22
)1(
4
)1(
2222
)1(
2*)1(2*)1()('
x
x
x
xxxx
x
xxxxxf
When the calculated derivative and NDER of the function on the calculator are graphed together. They appear to be identical, which is a strong indication that the calculation is correct.
Will be majoring in Biology at Rutgers University in the Fall.