Finding the Derivative of the Tangent Function Using the Quotient Rule

d/dx tan x

To find the derivative of the tangent function, we can use the quotient rule

To find the derivative of the tangent function, we can use the quotient rule. The tangent function is defined as tan(x) = sin(x)/cos(x).

Using the quotient rule, the derivative of tan(x) is given by:

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

Now, let’s find the derivatives of sin(x) and cos(x):

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

Substituting these values back into the derivative formula, we get:

d/dx tan(x) = (cos(x) * cos(x) – sin(x) * (-sin(x)))/(cos(x))^2
= (cos^2(x) + sin^2(x))/(cos^2(x))
= 1/(cos^2(x))

Since cos^2(x) is equivalent to (cos(x))^2, we can also write the derivative as:

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

Therefore, the derivative of tan x with respect to x is sec^2(x), where sec(x) represents the secant function.

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