Differentiation of tan x using first principle method
To find the derivative of tan(x) using the first principle method, we need to find the limit as h approaches 0 of the difference quotient:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
In this case, f(x) = tan(x).
Replacing f(x) and f(x + h) into the difference quotient:
f'(x) = lim(h->0) [tan(x + h) - tan(x)] / h
Next, we will use the trigonometric identity: tan(A) - tan(B) = sin(A - B) / cos(A) * cos(B). Applying this identity, we get:
f'(x) = lim(h->0) [sin(x + h - x) / (cos(x + h) * cos(x)) / h
Re-arranging the terms in the numerator, we have:
f'(x) = lim(h->0) [sin(h) / (cos(x + h) * cos(x)) / h
Now, we split the fraction into two separate parts:
f'(x) = lim(h->0) sin(h) / h * 1 / (cos(x + h) * cos(x))
The first part of the fraction, sin(h) / h, is a well-known limit that approaches 1 as h approaches 0. So, let's evaluate the second part of the fraction:
f'(x) = 1 * 1 / (cos(x + h) * cos(x))
Now, we apply the limit:
f'(x) = 1 / (cos(x) * cos(x))
Finally, simplifying the expression:
f'(x) = 1 / cos^2(x)
Therefore, the derivative of tan(x) using the first principle is 1 / cos^2(x) or sec^2(x).