The Derivative of the Tangent Function | Using the Quotient Rule to Find the Derivative of tan(x)

(d/dx) tan(x)

To find the derivative of the tangent function, we can use the quotient rule

To find the derivative of the tangent function, we can use the quotient rule.

The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:

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

In the case of f(x) = tan(x), we can write it as f(x) = sin(x)/cos(x).

Now, let’s find the derivatives of sin(x) and cos(x) separately to use in the quotient rule:

dsin(x)/dx = cos(x)
dcos(x)/dx = -sin(x)

Applying the quotient rule, we have:

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)

Recall that the identity cos^2(x) + sin^2(x) = 1. Using this identity, we can simplify the expression to:

f'(x) = 1 / cos^2(x)
= sec^2(x)

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

In summary, (d/dx) [tan(x)] = sec^2(x).

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