How To Derive The Derivative Of Tan(X) Using The Quotient Rule: A Step-By-Step Guide

derivative of tanx

sec^2x

The derivative of tan(x) is sec²(x).

To derive this, we use the quotient rule which states that the derivative of a function that is a quotient of two other functions f(x) and g(x) is equal to [(g(x)) * (f'(x)) – (f(x)) * (g'(x))] / [(g(x))^2].

With tan(x), we can rewrite it as sin(x) / cos(x).

So using the quotient rule, we get:

[(cos(x)) * (cos(x)) – (sin(x)) * (-sin(x))] / [(cos(x))^2]

Simplifying, we get:

(cos²(x) + sin²(x)) / (cos²(x))

Recall that cos²(x) + sin²(x) is equal to 1, so we get:

1 / (cos²(x))

Finally, we remember that sec(x) is equal to 1/cos(x), so we can write the derivative as:

(sec²(x))

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