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The first principles
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- 12
Differentiation from the first principles
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Definition of a derivative:What is a derivative?
The derivative of a function measures the rate at which the dependent variable changes as the
independent variable changes.
Qué?
In the Cartesian plane we use 𝑦 to represent the dependent variable, and 𝑥 to represent the
independent variable.
So, the derivative of a function measures the rate at which 𝒚 changes as 𝒙 changes.
In case you feel like you are suffering from a touch of déjà vu… “the rate at which 𝑦 changes as
𝑥 changes” we are indeed familiar with; we call it the gradient of a function.
In a Nutshell: The value of a derivative at a particular point is the same as the gradient of the
tangent to the curve at that point.
∴ 𝒎 = 𝒇′(𝒙)
𝑓′(𝑥) is one of the notations we use for the derivative… this reads as “the derivative of 𝑓 with
respect to 𝑥” or “𝑓 prime of 𝑥”, and is often referred to informally as “f dash”.
And we just so happen to have an equation for the gradient of the tangent to a curve:
𝑚𝑎𝑣𝑒
𝑓(𝑥+∆𝑥)−𝑓(𝑥) where ∆𝑥 → 0
= ∆𝑥
First thing we are going to do is make a simple substitution. Instead of ∆𝒙 being the change in the
𝑥- value, now we are going to represent the 𝑥-increment between the two points with a simple
variable instead… we will call it 𝒉… this is done for two reasons:
1. It is international standard practice
2. It will vastly simplify the appearance of the algebra. (you’ll thank me for this…)
So now our equation for the average gradient at a point on a curve is:
𝑚𝑎𝑣𝑒 𝑓(𝑥+ℎ)−𝑓(𝑥)
= ℎ where ℎ → 0
ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
As the value of ℎ → 0, the gradient of the resultant secant better approximates the gradient of the
1
, tangent to the curve at point A.
2
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