Derivative of Tangent Function: A Complete Guide with Step-by-Step Explanation

d/dx (tan x)

To find the derivative of the tangent function, we can use basic differentiation rules

To find the derivative of the tangent function, we can use basic differentiation rules. The derivative of the tangent function can be determined using the quotient rule.

The quotient rule states that if we have a function in the form f(x)/g(x), then the derivative is given by:
d/dx (f(x)/g(x)) = (g(x) * f'(x) – f(x) * g'(x))/[g(x)]^2

Using this rule, let’s find the derivative of the tangent function (tan x).

Let’s rewrite the tangent function as the ratio of the sine function (sin x) divided by the cosine function (cos x):
tan x = sin x / cos x

Now, let f(x) = sin x and g(x) = cos x.

Taking the derivatives, we have:
f'(x) = d/dx (sin x) = cos x [Using the derivative of the sine function, which gives us the cosine]
and
g'(x) = d/dx (cos x) = -sin x [Using the derivative of the cosine function, which gives us the negative sine]

Applying the quotient rule, we get:
d/dx (tan x) = [(cos x)(cos x) – (sin x)(-sin x)] / [cos x]^2

Simplifying the numerator, we have:
= cos^2(x) + sin^2(x)

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify further:
= 1

Therefore, the derivative of the tangent function (tan x) is equal to 1.

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

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