The Quotient Rule: Finding the Derivative of tan(x)

Derivitive of tan(x)

To find the derivative of tan(x), we will use the quotient rule

To find the derivative of tan(x), we will use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x))/[h(x)]^2

In the case of tan(x), we can rewrite it as the quotient of sin(x) and cos(x):

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

Now, we can find the derivatives of sin(x) and cos(x) using the chain rule:

d/dx(sin(x)) = cos(x)

d/dx(cos(x)) = -sin(x)

Applying the quotient rule to tan(x), we have:

tan'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x)))/[cos(x)]^2

Simplifying, we get:

tan'(x) = (cos^2(x) + sin^2(x))/[cos^2(x)]

Since sin^2(x) + cos^2(x) = 1 (from the Pythagorean identity), we have:

tan'(x) = 1/[cos^2(x)]

Recalling that tan(x) = sin(x)/cos(x), we can rewrite the result as:

tan'(x) = 1/cos^2(x)

Alternatively, we can express the derivative in terms of the secant function:

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

Therefore, the derivative of tan(x) is 1/cos^2(x) or equivalently sec^2(x).

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