Exploring the Derivative of Tan x | Trigonometric Definition and Sec^2 x Connection

The derivative of tan x is…

The derivative of tan x is sec^2 x, where sec x is the secant function

The derivative of tan x is sec^2 x, where sec x is the secant function.

To understand why the derivative of tan x is sec^2 x, we need to look at the trigonometric definition of these functions.

The tangent function, tan x, is defined as the ratio of the sine function to the cosine function: tan x = sin x / cos x.

The secant function, sec x, is the multiplicative inverse of the cosine function: sec x = 1 / cos x.

To find the derivative of tan x, we can express it in terms of sine and cosine:

tan x = sin x / cos x

Now, using the quotient rule of differentiation, we can find the derivative of tan x:

dy/dx = (d/dx)(sin x / cos x)
= [(cos x)(cos x) – (sin x)(-sin x)] / (cos x)^2 [Applying the quotient rule]
= (cos^2 x + sin^2 x) / cos^2 x
= 1 / cos^2 x
= sec^2 x

Therefore, the derivative of tan x is sec^2 x.

It is important to note that the derivative of tan x is not defined at points where cos x = 0, as division by zero is undefined. These points correspond to x-values where tan x has vertical asymptotes in the graph.

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