Question
(a) Use the method of first principles to determine the derivative of
(b) Use an appropriate method of differentiation to determine the derivative of the following functions (sim-
plify your answers as far as pnssible):
(i)
(ii)
(iii)
(b) Use an appropriate method of differentiation to determine the derivative of the following functions (sim-
plify your answers as far as pnssible):
(i)
(ii)
(iii)
Ask by Carlson Wagner. in South Africa
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Derivatives:
-
(a)
-
(b)(i)
-
(b)(ii)
-
(b)(iii)
Solution
Certainly! Let’s tackle each part of your question step by step.
Problem (a):
Use the method of first principles to determine the derivative of
.
Solution:
The derivative of a function
using first principles (the definition of the derivative) is given by:
Let’s apply this to
:
- Compute
:
- Find the difference
:
- Simplify the difference:
- Divide by
and take the limit as :
Answer:
Problem (b)(i):
Differentiate
.
Solution:
To find the derivative of a composite function, we’ll use the chain rule multiple times. Let’s break it down step by step.
-
Identify the innermost function and work outward:
- Let
- Then
- Next,
- Finally,
- Let
-
Differentiate each layer using the chain rule:
-
(Derivative ofis ) -
-
-
-
-
Apply the chain rule by multiplying the derivatives:
Substituting the derivatives:
- Simplify the expression:
Answer:
Problem (b)(ii):
Differentiate
.
Solution:
We’ll use the quotient rule for differentiation, which states:
For
:
-
Identify
and : -
-
Find the derivatives
and : -
-
Apply the quotient rule:
- Simplify the numerator:
Recall that
, so:
- Final expression for the derivative:
Answer:
Problem (b)(iii):
Differentiate
.
Solution:
First, let’s simplify the expression before differentiating.
- Simplify
:
- Differentiate
:
- Differentiate
:
Using the chain rule:
- Combine the results:
Answer:
Summary of Answers:
-
(a)
-
(b)(i)
-
(b)(ii)
-
(b)(iii)
Answered by UpStudy AI and reviewed by a Professional Tutor
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Bonus Knowledge
To find the derivative of
using first principles, we start with the definition of the derivative:
This simplifies to:
Thus, the derivative is
.
Now, for the derivatives of the following functions:
(i) For
, we use the chain rule. The derivative is:
(ii) For
, using the quotient rule:
(iii) For
, we simplify first:
The derivative is: