Answer:
Step-by-step explanation:
Given the function f(x) = tan (x)
From trigonometry identity, tan (x) = sin(x)/cos(x)
f(x) = sin(x)/cos(x)
Using the quotient rule to find the derivative of the function, we will have;
f'(x) = cos (x)cos (x) - sin (x)[-sin (x)]/ cosĀ²x
f'(x) = cosĀ²x - sinĀ²x/ cosĀ²x
Divide through by cosĀ²x
f'(x) = cosĀ²x/cosĀ²x + sinĀ²x/cosĀ²x / cosĀ²x/cosĀ²x
f'(x) = 1 + sinĀ²x/cosĀ²x / 1
f'(x) = 1 + (sinĀ²x/cosĀ²x)
f'(x) = 1 + tanĀ²x
From trig identity, Ā 1 + tanĀ²x = secĀ²x
Hence, f'(x) = secĀ²x
Hence the expression Ā f'(x) Ā in terms of the secant function is secĀ²x
f'(x) = 1/cosĀ²x
For the function to be defined, it means that cosĀ²x ā 0
cosĀ²x ā 0
cos x ā 0
x ā cosā»Ā¹0
x ā 90ā°
Hence the value of x must not be equal to 90 for the function to be defined. x can be defined at when x = 0 since cos 0 = 1, the equation will Ā becomes f'(0) = 1/cosĀ²0 = 1/1 = 1.
Hence the function is defined at 0ā¤x<90
