derivative of tanx
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 if we have a function in the form of 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
In this case, g(x) = sin(x) and h(x) = cos(x). Therefore, we have:
tan(x) = sin(x) / cos(x)
Now let’s find the derivative d(tan(x))/dx using the quotient rule:
[tan(x)]’ = [sin(x) / cos(x)]’
Using the quotient rule, we have:
[tan(x)]’ = [sin(x) * cos(x) – sin(x) * (-sin(x))] / (cos(x))^2
Simplifying this expression, we get:
[tan(x)]’ = [sin(x) * cos(x) + sin^2(x)] / (cos(x))^2
Now, we can simplify further by using the trigonometric identity:
sin^2(x) + cos^2(x) = 1
[tan(x)]’ = [sin(x) * cos(x) + 1 – cos^2(x)] / (cos(x))^2
Simplifying, we get:
[tan(x)]’ = sin(x) / cos^2(x)
Finally, we can rewrite this in terms of the secant function, which is the reciprocal of the cosine function:
[tan(x)]’ = sin(x) * sec^2(x)
Therefore, the derivative of tan(x) is sin(x) * sec^2(x).
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