Derivative of tan(x) – Applying the Quotient Rule and re-writing in terms of sec(x)

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 g(x) = u(x)/v(x), where u(x) and v(x) are differentiable functions, then the derivative of g(x) is given by:

g'(x) = (v(x) * u'(x) – u(x) * v'(x))/(v(x))^2

In the case of f(x) = tan(x), we can represent it as f(x) = sin(x)/cos(x), where u(x) = sin(x) and v(x) = cos(x).

Now, let’s take the derivatives of u(x) and v(x) to apply the quotient rule:

u'(x) = d/dx[sin(x)] [Using the chain rule]
= cos(x)

v'(x) = d/dx[cos(x)] [Using the chain rule]
= -sin(x)

Now we can substitute these values into the quotient rule formula:

f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x)))/(cos(x))^2
= (cos^2(x) + sin^2(x))/(cos^2(x))
= 1/cos^2(x)

However, we can express this derivative in terms of the secant function. Recall that sec(x) = 1/cos(x). Rewriting the derivative, we have:

f'(x) = sec^2(x)

Therefore, the derivative of f(x) = tan(x) with respect to x is f'(x) = sec^2(x).

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