Derivative of 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, denoted as tan(x), is defined as the ratio of the sine of x to the cosine of x:
tan(x) = sin(x) / cos(x)
Now, let’s find the derivative of tan(x) using the quotient rule:
The quotient rule states that for a function f(x) = g(x) / h(x), the derivative is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
For tan(x), g(x) = sin(x) and h(x) = cos(x).
Taking the derivatives of g(x) and h(x):
g'(x) = cos(x)
h'(x) = -sin(x)
Now, substituting these values into the quotient rule formula:
tan'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2
Simplifying further:
tan'(x) = (cos^2(x) + sin^2(x)) / [cos(x)]^2
Since, cos^2(x) + sin^2(x) = 1, we have:
tan'(x) = 1 / [cos(x)]^2
Therefore, the derivative of tan(x) is 1 / [cos(x)]^2.
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