Derivative of tan(x) – Quotient Rule Explained with Trigonometric Identity and Step-by-Step Examples

(d/dx) tan(x)

To find the derivative of tan(x), we can use the quotient rule

To find the derivative of tan(x), we can use the quotient rule. The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient u(x)/v(x) is given by:

(d/dx) (u(x)/v(x)) = (v(x) * d/dx[u(x)] – u(x) * d/dx[v(x)]) / (v(x))^2

In this case, u(x) = sin(x) and v(x) = cos(x), since tan(x) is equal to sin(x)/cos(x). Now, let’s find the derivatives of u(x) and v(x):

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

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

Now, substitute these derivatives into the quotient rule:

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

Simplifying further:

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

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

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

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

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