Discovering The Derivative Of Tanx: A Step-By-Step Guide

d/dx(secx)

secxtanx

To find the derivative of sec(x), we can use the formula:

d/dx(sec(x)) = sec(x) tan(x)

This formula can be derived using the chain rule and the fact that sec(x) = 1/cos(x).

To see why this formula is true, we can start by writing:

sec(x) = 1/cos(x)

Taking the derivative of both sides with respect to x using the quotient rule, we get:

d/dx(sec(x)) = d/dx(1/cos(x)) = -1/cos^2(x) * (-sin(x))

Simplifying this expression, we get:

d/dx(sec(x)) = sin(x) / cos^2(x)

Recalling that tan(x) = sin(x)/cos(x), we can rewrite this expression as:

d/dx(sec(x)) = (sin(x)/cos(x)) * (1/cos(x)) = sec(x) * tan(x)

Therefore, the derivative of sec(x) is sec(x) times tan(x).

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