Pregunta
QUESTION 4
Use the method of first principles to determine the derivative of
Use an appropriate method off differentiation to determine the derivative of the following functions (sim-
plify your answers as far as possible):
(i)
(ii)
(iii)
Use the method of first principles to determine the derivative of
Use an appropriate method off differentiation to determine the derivative of the following functions (sim-
plify your answers as far as possible):
(i)
(ii)
(iii)
Ask by Weber Fowler. in South Africa
Jan 23,2025
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Beyond the Answer
To find the derivative of
using first principles, we start with the definition of the derivative:
Calculating
:
Now, substituting into our limit gives us:
Canceling
(assuming
), we simplify to:
Taking the limit as
yields:
Next, we’ll use differentiation rules to find the derivatives of the given functions.
(i) For
:
The derivative is:
The derivative is:
(ii) For
:
Using the product rule:
Using the product rule:
This simplifies to:
(iii) For
:
Using logarithmic differentiation:
First, take the natural log:
Using logarithmic differentiation:
First, take the natural log:
Differentiate using product and chain rules:
Thus,
This expression can be simplified further if needed.

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