Understanding the Derivative | How to Find the Derivative of the Tangent Function using the Quotient Rule

Derivative of:tan(x)

The derivative of the tangent function, denoted as tan(x), can be found using the quotient rule

The derivative of the tangent function, denoted as tan(x), can be found using the quotient rule. The quotient rule states that for a function u(x)/v(x), where u(x) and v(x) are differentiable functions of x,

(d/dx)(u(x)/v(x)) = [v(x)(d/dx)(u(x)) – u(x)(d/dx)(v(x))]/[v(x)]^2

Applying the quotient rule to tan(x) = sin(x)/cos(x), where sin(x) represents the sine function and cos(x) represents the cosine function, we can differentiate both the numerator and denominator separately.

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

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x), so we can substitute these values into the equation:

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

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

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

= 1 / cos^2(x)

The reciprocal of cos^2(x) is sec^2(x), which is the square of the secant function, so the final derivative is:

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

Therefore, the derivative of the tangent function tan(x) is sec^2(x).

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