The Quotient Rule: How to Differentiate Tan(x) with Respect to x?

d/dx(tanx)

To differentiate the function tan(x) with respect to x, we can use the quotient rule

To differentiate the function tan(x) with respect to x, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative can be found using the formula:

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

Let’s apply the quotient rule to differentiate tan(x):

We have g(x) = sin(x) and h(x) = cos(x).

Taking the derivatives of g(x) and h(x), we get:

g'(x) = cos(x) (derivative of sin(x))
h'(x) = -sin(x) (derivative of cos(x))

Plugging these values into the quotient rule formula, we have:

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

Simplifying the numerator, we have:

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

Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we can simplify further:

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

Since tan(x) is equivalent to sin(x) / cos(x), we can rewrite this as:

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

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

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