Question
- What is the geometrical meaning of differentiation?
- Differentiate the following functions using the first principle.
(a)
(b)
©
(d)
(a)
(b)
©
(d)
Ask by Hampton Harris. in Zambia
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Differentiation represents the slope of the tangent to a curve at a point, indicating the instantaneous rate of change of the function.
(a)
:
(b)
:
©
:
(d)
:
Solution
(a)
Let
. Then
Expanding,
Thus,
Taking the limit as
,
(b)
Let
. Then
The difference is
Combine the fractions:
We expand both products:
- Expand
:
- Expand
:
Subtracting the second expansion from the first:
Thus,
Taking the limit as
, note that
. Hence,
©
Let
. Then
We have
To simplify, multiply the numerator and denominator by the conjugate:
Simplifying the numerator:
So,
Taking the limit as
,
(d)
Since this is a constant function, we have
Thus,
Then,
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Mind Expander
Differentiation geometrically represents the slope of the tangent line to a curve at a given point. Picture a roller coaster: the derivative tells you how steep the ride is at any point, allowing you to understand how fast the elevation is changing. So, as you’re tangling with functions, think of differentiation as your trusty slope calculator for curves!
To differentiate the given functions using the first principle, you can apply the formula:
For
:
- Substitute to find
. - Simplify
. - Take the limit as
approaches 0 to find .
Repeat this process for the other functions:
-
yields . -
gives . - For the constant function
, the derivative is zero since a constant has no slope.
Happy differentiating!