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 tangent function is defined as “tan(x) = sin(x) / cos(x)”. Let’s differentiate both the numerator and denominator separately and then apply the quotient rule.

First, let’s differentiate the numerator:
(d/dx) sin(x)

The derivative of the sine function is the cosine function:
(d/dx) sin(x) = cos(x)

Next, let’s differentiate the denominator:
(d/dx) cos(x)

The derivative of the cosine function is the negative sine function:
(d/dx) cos(x) = -sin(x)

Now, let’s use the quotient rule to find the derivative of tan(x):
(d/dx) tan(x) = [(d/dx) sin(x) * cos(x) – sin(x) * (d/dx) cos(x)] / [cos(x)]^2

Substituting the values we found earlier:
(d/dx) tan(x) = [cos(x) * cos(x) – sin(x) * (-sin(x))] / [cos(x)]^2

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

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:
(d/dx) tan(x) = 1 / [cos(x)]^2

Therefore, the derivative of tan(x) is equal to 1 divided by the square of the cosine function.

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