1st Derivative Test

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Definition: - If x be a point in the domain of definition of a function f(x) or y

[let y = f(x)] and to an increment ?x of x, there is a corresponding increase in ?y, then the first derivative

test is
            f’(a) = [f (a + h) – f(a)] / h

Where h = ? x and the derivative of f(x) with aspect to x at x = a and denoted by f (a) and f ’(a) is the

derivative if f (a).


 
Example 1: - Find the derivatives of the following with respect to x.

i)             y = x ^3

ii)            y = x + 2

iii)           y = 7 / x


 
Solution: -

i)   y = x^3

Differentiating both sides with respect to x, then we get

            dy / dx = d / dx (x^3)

            dy / dx = 3* x ^ (3 – 1)

            dy / dx = 3* x^2

Therefore dy / dx = 3x^2

 
ii)  y = x + 2

dy / dx = d / dx ( x+ 2)

dy / dx = d / dx (x) + d / dx (2)

dy / dx = 1 + 0

Therefore dy / dx = 1

 
iii) Y = 7 / x

Dy / dx = d / dx ( 7 / x)

Dy / dx = 7 * d / dx ( 1/ x)

Dy / dx = 7 * log x.

Therefore dy / dx = 7 log x

 

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