A Complete Guide to Finding the Derivative of the Tangent Function

Derivative of tan(x)

The derivative of the tangent function, denoted as tan(x), can be found using standard differentiation rules

The derivative of the tangent function, denoted as tan(x), can be found using standard differentiation rules.

Let’s begin by recalling the basic trigonometric identity:

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

Now, we can differentiate both sides of this equation with respect to x.

To differentiate the right side, we can use the quotient rule, which states that if we have a function u(x) divided by another function v(x), then the derivative of the quotient is given by:

(u/v)’ = (u’v – uv’) / v^2

Applying the quotient rule to sin(x) / cos(x), we have:

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

Now, let’s find the derivatives of sin(x) and cos(x) separately:

d/dx(sin(x)) = cos(x) (derivative of sin(x) is cos(x))

d/dx(cos(x)) = -sin(x) (derivative of cos(x) is -sin(x))

Substituting these derivatives into the quotient rule formula, we get:

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

Simplifying further:

d/dx [sin(x) / cos(x)] = [cos^2(x) + sin^2(x)] / cos^2(x)

Now, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify the expression:

d/dx [sin(x) / cos(x)] = 1 / cos^2(x)

Since tan(x) = sin(x) / cos(x), our final result is:

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

Alternatively, we can express the derivative of tan(x) in terms of the secant function:

d/dx [tan(x)] = sec^2(x)

Both expressions are equivalent and represent the derivative of the tangent function.

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