The Process of Reversing Differentiation
undefined
LO: To find the function knowing the
derivative of the function.
Reversing the process of
differentiation
7 October 2024
When differentiating powers of
x
,
y
=
x
n
, we multiply by the
original exponent
and then drop the exponent by one.
We can write this process as follows:
nx
n-
1
Reversing the process of differentiation given above would
give
nx
n
x
n
Reversing the process of differentiation
Reversing the process of differentiation
For example,
2
x
We therefore have to write
y
=
x
2
+
c
.
This set, or family, of all anti-derivatives of a function is
called the indefinite integral
The process of finding a function
f
(
x
)
whose derivative
is
f
’(
x
)
is called anti-differentiation, which relates to
integration
.
What is the derivative of
x
2
with respect to
x
?
What is the derivative of
x
2
+ 3
with respect to
x
?
2
x
What is the derivative of
x
2
– 5
with respect to
x
?
2
x
If you anti-differentiate
2
x
with respect to
x
x
2
You can easily see that
2
x
is the derivative for any
function of the form
x
2
+ c,
where
c
is any real number
The
antiderivative
of all of them is the same
Reversing the process of differentiation
General form of an Antiderivative
Let
F
be an antiderivative of
f
over an interval
I
, then
ii. If
G
is an antiderivative of
f
over
I
,
there is a
constant
C
to which
G
(
x
) =
F
(
x
) +
C
over
I
.
i. For each constant
C
, the function
F
(
x
) +
C
is also
an antiderivative of
f
over
I
;
In other words, the most general form of the
antiderivative of
f
over
I
is
F
(
x
) +
C
For example:
Adding 1 to the power and dividing by the new power gives:
= 2
x
3
This is not the complete solution, however, because if we
differentiated
y
= 2
x
3
+ 1,
We therefore have to write
y
= 2
x
3
+
c
.
or
y
= 2
x
3
– 3,
or
y
= 2
x
3
+
any
constant
In general:
The antiderivative of
x
n
are given by
Reversing the process of differentiation
Find the antiderivative of the function
x
10
:
Adding 1 to the power and dividing by the new power gives:
Adding 1 to the power and dividing by the new power gives:
= x
-5
Reversing the process of differentiation
Adding 1 to the power and dividing by the new power gives:
Rewriting using rational exponents
Reversing the process of differentiation
Antidifferentiation is also known as
indefinite integration
If
F’(x) = f(x),
we write
And is denoted with an integral symbol,
integrand
Variable of integration
Constant of integration
Is read as:
“The antiderivative of
f
with respect to
x
”
“The integral of
f
with respect to
x
”
Reversing the process of differentiation
Rules to find the indefinite integrals
Power rule
Constant rule
Constant multiple rule
Sum or difference rule
Indefinite integrals
Find the indefinite integral
Applying the power rule
Find the indefinite integral
Applying the constant rule
The
dt
tells you that the variable of integration is
t
Find the indefinite integral
Applying the constant multiple rule
Indefinite integrals
Find the indefinite integral
Applying the sum or difference rule
Applying the constant
multiple rule
Applying the power rule
and constant rule
Indefinite integrals
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Reversing the differentiation process involves finding the function knowing its derivative. By increasing the exponent by 1, dividing by the new exponent, and adding a constant term, the antiderivative of a function can be determined. The general form of an antiderivative involves adding a constant, making it flexible for various functions. Examples and explanations of antiderivatives are provided to enhance understanding and application in calculus.
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Presentation Transcript
7 October 2024 Reversing the process of differentiation LO: To find the function knowing the derivative of the function. www.mathssupport.org
Reversing the process of differentiation When differentiating powers of x, y = xn, we multiply by the original exponent and then drop the exponent by one. We can write this process as follows: nxn reduce the exponent by 1 multiply by the the original exponent xn nxn-1 Reversing the process of differentiation given above would give nxn xn increase the exponent by 1 divide by the exponent nxn-1 dy dx Suppose we are given the derivative = xnand asked to find y in terms of x. increase the exponent by 1 xn+1 ??+1 ? + 1 divide by the exponent xn www.mathssupport.org
Reversing the process of differentiation The process of finding a function f(x) whose derivative is f (x) is called anti-differentiation, which relates to integration. For example, What is the derivative of x2 with respect to x? If you anti-differentiate 2xwith respect to x 2x x2 What is the derivative of x2 + 3with respect to x? What is the derivative of x2 5with respect to x? You can easily see that 2xis the derivative for any function of the formx2 + c, where c is any real number The antiderivative of all of them is the same 2x 2x This set, or family, of all anti-derivatives of a function is called the indefinite integral We therefore have to write y = x2 + c. www.mathssupport.org
Reversing the process of differentiation General form of an Antiderivative Let Fbe an antiderivative of f over an interval I, then i. For each constant C, the function F(x) + Cis also an antiderivative of f over I; ii. If G is an antiderivative of foverI, there is a constant C to which G(x) = F(x) + Cover I. In other words, the most general form of the antiderivative of fover I is F(x) + C www.mathssupport.org
Reversing the process of differentiation dy dx For example: Adding 1 to the power and dividing by the new power gives: 2 If =6 find in terms of . y x x 6 3 =3 = 2x3 y x This is not the complete solution, however, because if we differentiated y = 2x3 + 1, ory = 2x3 3, or y = 2x3 + any constant dy x dx we would also get 2 = 6 We therefore have to write y = 2x3 + c. In general: The antiderivative of xnare given by 1 ? + 1??+1+ ? www.mathssupport.org
Reversing the process of differentiation Find the antiderivative of the function x10: Adding 1 to the power and dividing by the new power gives: 1 10 + 1?10+1+ ? 1 11?11+ ? = Find the antiderivative of the function 1 Adding 1 to the power and dividing by the new power gives: 1 5 + 1? 5+1+ ? 1 4? 4+ ? = x-5 ?5 1 = 4?4+ ? = www.mathssupport.org
Reversing the process of differentiation Find the antiderivative of the function Rewriting using rational exponents 4?3 3 4 = ? Adding 1 to the power and dividing by the new power gives: 1 3 4+ 1 =1 7 4 =4 7? 3 4+1+ ? ? 7 4+ ? ? 7 4+ ? www.mathssupport.org
Reversing the process of differentiation Antidifferentiation is also known as indefinite integration And is denoted with an integral symbol, ?? IfF (x) = f(x), we write ? ? ?? = ? ? + ? Constant of integration integrand Variable of integration Is read as: The antiderivative of fwith respect to x The integral of fwith respect to x ? ? ?? www.mathssupport.org
Rules to find the indefinite integrals Power rule 1 ? + 1??+1+ ?,? 1 ???? = Constant rule ??? = ?? + ?, Constant multiple rule ??(?)?? = ? ?(?)?? Sum or difference rule = ? ? ?? ? ? ?? ?(?) ?(?)?? www.mathssupport.org
Indefinite integrals Find the indefinite integral Applying the power rule ?6?? 1 ? + 1??+1+ ?,? 1 ???? = 1 6 + 1?6+1+ ? =1 7?7+ ? Find the indefinite integral Applying the constant rule 4?? ??? = ?? + ?, Thedttells you that the variable of integration is t = 4? + ?, www.mathssupport.org
Indefinite integrals Find the indefinite integral Applying the constant multiple rule 3?5?? ??(?)?? = ? ?(?)?? = 3 ?5?? 1 5 + 1?5+1+ ? = 3 =1 2?6+ ? www.mathssupport.org
Indefinite integrals Find the indefinite integral Applying the sum or difference rule (3?4+6?2+ 2)?? ?(?) ?(?)?? = ? ? ?? ? ? ?? = 3?4?? + 6?2?? + 2?? Applying the constant multiple rule = 3 ?4?? +6 ?2?? + 2?? 1 1 Applying the power rule and constant rule 4 + 1?4+1 2 + 1?2+1 = 3 +6 +2? + ? =3 5?5+ 2?3+ 2? + ? www.mathssupport.org
Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org
