Question
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(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)

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 :
  1. Compute :
  1. Find the difference :
  1. Simplify the difference:
  1. 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.
  1. Identify the innermost function and work outward:
    • Let
    • Then
    • Next,
    • Finally,
  2. Differentiate each layer using the chain rule:

    • (Derivative of is )
  3. Apply the chain rule by multiplying the derivatives:
Substituting the derivatives:
  1. Simplify the expression:
Answer:

Problem (b)(ii):

Differentiate .
Solution:
We’ll use the quotient rule for differentiation, which states:
For :
  1. Identify and :
  2. Find the derivatives and :
  3. Apply the quotient rule:
  1. Simplify the numerator:
Recall that , so:
  1. Final expression for the derivative:
Answer:

Problem (b)(iii):

Differentiate .
Solution:
First, let’s simplify the expression before differentiating.
  1. Simplify :
  1. Differentiate :
  1. Differentiate :
Using the chain rule:
  1. 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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Explain
Simplify this solution

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:

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