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The Difference Rule: [f(x) − g(x)]
= f
(x) − g
(x).
Derivative of the Natural Exponential Function: (e
x
)
= e
x
.
The Product Rule: [f(x)g(x)]
= f
(x)g(x)+f(x)g
(x).
The Quotient Rule:
f(x)
g(x)
=
f
(x)g(x) − f(x)g
(x)
g
2
(x)
.
Derivatives of Trigonometric Functions:
(sin x)
=cosx, (cos x)
= −sin x,
(tan x)
=sec
2
x, (cot x)
= −csc
2
x,
(csc x)
= −csc x cot x, (sec x)
=secx tan x.
4.7. The Chain Rule
If f(x) and g(x) are both differentiable and h(x)=f(g(x)), then h
is differentiable and h
is given by the product
h
(x)=f
(g(x)) × g
(x).
4.8. Implicit Differentiation
Some functions are defined implicitly by a relation between x and y
such as
x
2
+ y
2
=25 (1)
or
x
3
+ y
3
=6xy. (2)
In some cases it is possible to solve such an equation for y as an
explicit function (or several functions) of x. For instance, if we solve
(1) for y,wegety = ±
√
25 − x
2
so two functions determined by the
implicit (1) are f(x)=
√
25 − x
2
and g(x)=−
√
25 − x
2
.
It’s not easy to solve (2) for y explicitly as a function of x by hand.
Nonetheless, 2 is the equation of a certain curve and it implicitly
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