Master The Quotient Rule: Derivative Of The Tangent Function As Sec²X

derivative of tanx

sec^2x

The derivative of the tangent function is given by:

d/dx (tan x) = sec²x

To derive this, we need to use the quotient rule for differentiation. Recall that the tangent function is defined as:

tan x = sin x / cos x

To apply the quotient rule, we need to find the derivative of the numerator and denominator separately. Using the product rule, we have:

d/dx (sin x) = cos x

d/dx (cos x) = -sin x

Therefore, the derivative of the numerator is cos x and the derivative of the denominator is -sin x. Now we can apply the quotient rule:

d/dx (tan x) = (cos x)(cos x) – (sin x)(-sin x) / cos²x

Simplifying this expression using trigonometric identities, we get:

d/dx (tan x) = sec²x

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