d/dx [tanx]
To find the derivative of the function f(x) = tan(x), we can use the quotient rule
To find the derivative of the function f(x) = tan(x), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then its derivative can be calculated as:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
Applying the quotient rule, we consider g(x) = sin(x) and h(x) = cos(x). Now, let’s find the derivatives of g(x) and h(x):
g'(x) = cos(x) (derivative of sin(x) is cos(x))
h'(x) = -sin(x) (derivative of cos(x) is -sin(x))
Now, substituting these values into the quotient rule:
f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2
f'(x) = (cos^2(x) + sin^2(x)) / cos^2(x)
By using the trigonometric identity cos^2(x) + sin^2(x) = 1, we have:
f'(x) = 1 / cos^2(x)
Since sec(x) is defined as the reciprocal of cos(x), we can rewrite the derivative as:
f'(x) = sec^2(x)
Therefore, the derivative of f(x) = tan(x) is f'(x) = sec^2(x).
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