Derivative of tan x
The derivative of the tangent function, written as d/dx(tan(x)), can be evaluated using trigonometric identities and differentiation rules
The derivative of the tangent function, written as d/dx(tan(x)), can be evaluated using trigonometric identities and differentiation rules.
To find the derivative of tan(x), we first express it in terms of sine and cosine using the identity tan(x) = sin(x) / cos(x).
Next, we apply the quotient rule of differentiation, which states that if f(x) = g(x) / h(x), then d/dx(f(x)) = (g'(x)h(x) – g(x)h'(x)) / (h(x))^2.
Applying the quotient rule to tan(x) = sin(x) / cos(x), we get:
d/dx(tan(x)) = (cos(x) * sin'(x) – sin(x) * cos'(x)) / (cos(x))^2
The derivative of sin(x) with respect to x is cos(x), and the derivative of cos(x) with respect to x is -sin(x).
Substituting these values into the equation, we have:
d/dx(tan(x)) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / (cos(x))^2
= (cos^2(x) + sin^2(x)) / (cos^2(x))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, the equation becomes:
d/dx(tan(x)) = 1 / (cos^2(x))
Since 1 / (cos^2(x)) is equivalent to sec^2(x), the derivative of tan(x) is:
d/dx(tan(x)) = sec^2(x)
Therefore, the derivative of tan(x) is equal to sec^2(x).
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