d/dx [tan x]
To find the derivative of the tangent function (tan x), we can use the quotient rule
To find the derivative of the tangent function (tan x), we can use the quotient rule.
The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) can be found using the formula:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
In this case, g(x) = sin x and h(x) = cos x.
Using the quotient rule, we can find the derivative of f(x) = tan x:
f'(x) = (cos x * cos x – sin x * (-sin x)) / (cos x)^2
= (cos^2(x) + sin^2(x)) / cos^2(x)
Now, recall the Pythagorean identity sin^2(x) + cos^2(x) = 1.
Therefore, f'(x) = 1 / cos^2(x)
Since sec^2(x) = 1 / cos^2(x), we can rewrite the derivative as:
f'(x) = sec^2(x)
Therefore, the derivative of tan x (d/dx [tan x]) is equal to sec^2(x).
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