Discover the Derivative of Tan(x) with Quotient Rule and Trigonometric Identities

(d/dx) tan(x)

sec^2(x)

The derivative of tan(x) with respect to x can be found using the quotient rule.

tan(x) = sin(x) / cos(x)

Using the quotient rule:

(d/dx) tan(x) = [(cos(x) * d/dx(sin(x))) – (sin(x) * d/dx(cos(x)))] / cos^2(x)

Now, we need to find the derivatives of sin(x) and cos(x).

(d/dx) sin(x) = cos(x)

(d/dx) cos(x) = -sin(x)

Substituting these back into the quotient rule formula, we get:

(d/dx) tan(x) = [(cos(x) * cos(x)) – (sin(x) * -sin(x))] / cos^2(x)

Simplifying, we get:

(d/dx) tan(x) = (cos^2(x) + sin^2(x)) / cos^2(x)

Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we get:

(d/dx) tan(x) = 1 / cos^2(x)

Therefore, the derivative of tan(x) with respect to x is 1 / cos^2(x).

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