𝑑/𝑑𝑥[tan 𝑥]
To find the derivative of tan(x), we can use the quotient rule
To find the derivative of tan(x), we can use 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
In the case of tan(x), we have f(x) = tan(x), and g(x) = sin(x) and h(x) = cos(x).
So we need to find g'(x) and h'(x) to plug into the quotient rule formula.
Using the derivative formulas, we know that the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
Therefore, g'(x) = cos(x) and h'(x) = -sin(x).
Plugging these values into the quotient rule formula, we get:
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
Using the identity sin^2(x) + cos^2(x) = 1, we can simplify the expression even further:
tan'(x) = 1 / [cos(x)]^2
Therefore, the derivative of tan(x) is 1 / [cos(x)]^2 or equivalently, sec^2(x).
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