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Topic 3: Differentiation (Derivatives) - Coggle Diagram
Topic 3:
Differentiation
(Derivatives)
Definition of Derivative
If f(x) is a function defined on an open interval I=(a,b) that contains the point c, the derivative of f at c, denoted by f'(c), is
provided the limit exists.
If f is differentiable at all
, f is differentiable on
I
. In this case, we can define a new function f'=f'(x) on
I
where f'(x) assumes the value of the derivative of f at
. We call f' the derivative of f, and write
.
Derivative Rules
Derivative of a Constant Function:
Power Rule:
if f(x)=
with
Constant Multiple Rule:
Sum Rule:
Product Rule:
Quotient Rule:
for x with
Derivative of Trigonometric Functions and Their Inverses
Derivatives of Logarithmic and Exponential Functions
For all x>0,
For all
, a is constant.
Chain Rule
If f=f(u) is differentiable at the point u=g(c), and g=g(x) is differentiable at the point x=c, then the composite function
is differentiable at x=c, and
In Leibniz's notation, if y=f(u) and u=g(x), then
Two important formulas of the above fact are:
where we have let f(u)=ln u
where we have let f(u)=
Higher Derivatives
For a differentiable function f=f(x),
first derivative of f: f'=f'(x)=
second derivative of f: f''=f''(x)=
n th derivative of f:
Implicit Differentiation
We apply implicit differentiation when a variable is defined implicitly as a function of another variable. For example, the equation
defines two functions of x, i.e. y=
,
. To find the derivative of y with respect to x, we do not need to solve y as a function of x. Instead, we use implicit differentiation. We treat y as a differentiable function of x and differentiate both sides of the equation with respect to x, then solve for
in terms of x and y together.
L'Hospital's Rule
Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval
I
containing a. Suppose also that
on
I
if
, then
if the limit on the right exists (or is
or
)
Application of Derivatives
Critical Points
Let f(x) be a function with domain
. A point
is called a critical point of f if either f'(c)=0 or f'(c) fails to exist.
The Second Derivative Test for Local Extrema
Let f(x) be differentiable on an interval I containing x=a, and f
'(x) also differentiable at x=a. Suppose that f'(a)=0.
(i). If f''(a)<0, then x=a is a local maximum
(ii). If f''(a)>0, then x=a is a local minimum
(iii). If f''(a)=0, then inconclusive
Steps to Find Extrema (Maximum/Minimum)
Step 1: Find critical points, x=a:
Find
, then let
to get critical point(s), x=a
Step 2: compute
Step 3: Determine max/min
Substitute all critical points into
, then check with the following:
i. If f''(a)<0, then x=a is a local maximum ii. If f''(a)>0, then x=a is a local minimum
iii. If f''(a)=0, then inconclusive
Step 4: Determine y values (if requested to find max/min points)
Substitute the critical point x=a into y. Then write the max/min point as (x,y).
NOTE: This step 4 is performed if the question requests y values.)
