tan(x) derivative
The derivative of the tangent function, tan(x), can be found using the quotient rule
The derivative of the tangent function, tan(x), can be found using the quotient rule. The quotient rule is a formula that allows us to find the derivative of a function that is the division of two functions. In the case of tan(x), it can be represented as the division of the sine function (sin(x)) and the cosine function (cos(x)), i.e., tan(x) = sin(x)/cos(x).
To find the derivative of tan(x), we will start by applying the quotient rule. Let’s denote f(x) = sin(x) as the numerator function and g(x) = cos(x) as the denominator function. The quotient rule states:
If h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) – f(x)g'(x))/[g(x)]^2.
Now, let’s determine the derivatives of f(x) and g(x).
The derivative of sin(x) is cos(x) (you can also remember it as d(sin(x))/dx = cos(x)).
The derivative of cos(x) is -sin(x) (you can also remember it as d(cos(x))/dx = -sin(x)).
Using the quotient rule and the derivatives of f(x) and g(x) that we just obtained, we can find the derivative of tan(x) as follows:
tan'(x) = [(cos(x) * cos(x)) – (sin(x) * (-sin(x))) ] / [cos(x)]^2
= (cos^2(x) + sin^2(x)) / [cos(x)]^2.
Since cos^2(x) + sin^2(x) equals 1 (due to the Pythagorean identity), we can simplify the expression further:
tan'(x) = 1 / [cos(x)]^2.
Hence, the derivative of tan(x) is given by tan'(x) = 1 / [cos(x)]^2.
Note: Sometimes in calculus, the derivative of the tangent function is also written as sec^2(x), where sec(x) represents the secant function. This is because the reciprocal of the cosine function is the secant function, and the derivative of the secant function is sec(x) * tan(x).
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