Mastering The Quotient Rule: Derivative Of Tan(X) = Sec^2(X)

What is the derivative of tan(x)?

sec^2(x)

The derivative of tan(x) is sec^2(x).

To obtain this result, we can use the quotient rule of differentiation, which says that for a function u(x) divided by v(x), the derivative is given by:

(u/v)’ = (u’v – uv’)/v^2

In this case, we have u(x) = sin(x) and v(x) = cos(x), so:

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

Using the quotient rule, we have:

tan'(x) = [(cos(x) * cos(x)) – (sin(x) * (-sin(x))))/(cos(x))^2

Simplifying this expression, we get:

tan'(x) = (cos^2(x) + sin^2(x))/(cos^2(x))

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

tan'(x) = 1/(cos^2(x))

Finally, using the identity sec^2(x) = 1/(cos^2(x)), we obtain:

tan'(x) = sec^2(x)

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