Understanding How to Find the Derivative of the Tangent Function Using the Quotient Rule

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 formula for the derivative of tan(x) with respect to x is:

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

Here, sec^2(x) represents the secant function squared.

The quotient rule states that if you have a function 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

Applying the quotient rule, we have:
d/dx(tan(x)) = (d/dx(sinx) * cosx – sinx * d/dx(cosx))/[cosx]^2

Now, let’s differentiate sin(x) and cos(x) individually:
d/dx(sinx) = cos(x)
d/dx(cosx) = -sin(x)

Substituting these derivatives into our previous equation, we get:
d/dx(tan(x)) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2

Simplifying further, we have:
d/dx(tan(x)) = (cos^2(x) + sin^2(x))/ [cos(x)]^2

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify it to:
d/dx(tan(x)) = 1 / [cos(x)]^2

Finally, we know that sec(x) = 1 / cos(x). Therefore, we can write the derivative of tan(x) as:
d/dx(tan(x)) = sec^2(x)

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

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