Derivative of the Tangent Function: A Step-by-Step Guide and Formula

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 quotient rule states that if you have a function of the form f(x) = g(x) / h(x), then the derivative of f(x) with respect to x is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In this case, the tangent function we are given is f(x) = tan(x), which can be written as f(x) = sin(x) / cos(x), using the trigonometric identity for tangent.

Using the quotient rule, we can find the derivative.

Let g(x) = sin(x) and h(x) = cos(x).

g'(x) = cos(x) (differentiate sin(x))
h'(x) = -sin(x) (differentiate cos(x))

Now we can substitute the values into the quotient rule formula:

f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / (cos(x))^2

Simplifying further:

f'(x) = (cos^2(x) + sin^2(x)) / (cos^2(x))

Using the Pythagorean identity: sin^2(x) + cos^2(x) = 1

f'(x) = 1 / (cos^2(x))

Therefore, the derivative of the tangent function, d/dx(tan(x)) is equal to 1 divided by the square of the cosine function: 1 / (cos^2(x)).

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

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