Derivative tanx
The derivative of tan(x) can be found using the trigonometric identity for tan(x), which states that tan(x) is equal to sin(x) divided by cos(x)
The derivative of tan(x) can be found using the trigonometric identity for tan(x), which states that tan(x) is equal to sin(x) divided by cos(x).
To differentiate tan(x) with respect to x, we can use the quotient rule. The quotient rule states that if you have a function f(x) = g(x) / h(x), the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
Let’s apply this rule to find the derivative of tan(x):
Using the trigonometric identity, we can rewrite tan(x) as sin(x) / cos(x).
Applying the quotient rule:
f'(x) = ([sin'(x) * cos(x)] – [sin(x) * cos'(x)]) / [cos(x)]^2
The derivative of sin(x) is simply cos(x), and the derivative of cos(x) is -sin(x). Substituting these values:
f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2
f'(x) = (cos^2(x) + sin^2(x)) / [cos(x)]^2
Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we can simplify further:
f'(x) = 1 / [cos(x)]^2
Finally, since tan(x) is defined as sin(x) / cos(x):
f'(x) = 1 / [cos(x)]^2 = sec^2(x)
Therefore, the derivative of tan(x) is sec^2(x).
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