The Quotient Rule: How to Find the Derivative of Tan(x)

d/dx [tanx]

To find the derivative of the function f(x) = tan(x), denoted as d/dx [tan(x)], we can use the quotient rule as follows:

f(x) = tan(x) = sin(x) / cos(x)

Using the quotient rule, the derivative is given by:

d/dx [tan(x)] = (cos(x) * d/dx[sin(x)] – sin(x) * d/dx[cos(x)]) / (cos(x))^2

We now need to find the derivatives of sin(x) and cos(x)

To find the derivative of the function f(x) = tan(x), denoted as d/dx [tan(x)], we can use the quotient rule as follows:

f(x) = tan(x) = sin(x) / cos(x)

Using the quotient rule, the derivative is given by:

d/dx [tan(x)] = (cos(x) * d/dx[sin(x)] – sin(x) * d/dx[cos(x)]) / (cos(x))^2

We now need to find the derivatives of sin(x) and cos(x).

d/dx [sin(x)] = cos(x) (using the derivative of sin(x), which is cos(x))
d/dx [cos(x)] = -sin(x) (using the derivative of cos(x), which is -sin(x))

Substituting these derivatives into the quotient rule expression, we have:

d/dx [tan(x)] = (cos(x) * cos(x) – sin(x) * (-sin(x))) / (cos(x))^2
= (cos^2(x) + sin^2(x)) / (cos(x))^2
= 1 / (cos(x))^2

Therefore, the derivative of tan(x) is d/dx [tan(x)] = 1 / (cos(x))^2.

Note that this derivative is also represented as sec^2(x), which is another form of the derivative of tan(x).

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