The Derivative of Tanx: Using the Quotient Rule and Simplifying using Trigonometric Identities

Derivative of tanx

The derivative of tanx can be found using the quotient rule

The derivative of tanx can be found using the quotient rule. The quotient rule states that for functions f(x) and g(x), the derivative of their quotient is given by:

(f(x)/g(x))’ = (f'(x)g(x) – f(x)g'(x))/(g(x))^2

In this case, f(x) is sinx and g(x) is cosx. So, we can write tanx as sinx/cosx.

Applying the quotient rule, we find:

(tanx)’ = ((sinx)'(cosx) – (sinx)(cosx)’)/((cosx)^2)

The derivative of sinx is cosx, and the derivative of cosx is -sinx. Substituting these values:

(tanx)’ = (cosx*cosx – sinx*(-sinx))/(cosx)^2

Simplifying further:

(tanx)’ = (cos^2x + sin^2x)/(cos^2x)

Using the trigonometric identity cos^2x + sin^2x = 1:

(tanx)’ = 1/(cos^2x)

We know that secx = 1/(cosx), so we can rewrite the result as:

(tanx)’ = sec^2x

Therefore, the derivative of tanx is sec^2x.

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