How to Find the Derivative of the Tangent Function | Step-by-Step Guide and Formula Explanation

d/dx tanx

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 derivative of the tangent function with respect to x, denoted as d/dx tan(x), can be calculated as follows:

Step 1: Recall that the tangent function can be expressed as tan(x) = sin(x)/cos(x).

Step 2: Apply 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) * u'(x) – u(x) * v'(x)) / [v(x)]^2.

In this case, u(x) = sin(x) and v(x) = cos(x). Differentiating u(x) and v(x) gives:

u'(x) = cos(x) and v'(x) = -sin(x).

Step 3: Substitute the values into the quotient rule formula:

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^2(x).

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

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

Alternatively, we can rewrite this in terms of secant (sec(x)) using the identity sec(x) = 1 / cos(x):

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

To summarize, the derivative of the tangent function with respect to x is equal to the square of the secant function, denoted as d/dx tan(x) = sec^2(x).

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