The Derivative of Tangent: Simplifying with the Quotient Rule and Trigonometric Identities

d tan(x)

The expression “d tan(x)” refers to the derivative of the trigonometric function tan(x) with respect to x

The expression “d tan(x)” refers to the derivative of the trigonometric function tan(x) with respect to x. To find the derivative of tangent, we can use the quotient rule. The quotient rule states that if you have a function of the form h(x) = f(x)/g(x), then the derivative is given by:

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

In this case, our function is tan(x), so f(x) = sin(x) and g(x) = cos(x).

Using the quotient rule, we can calculate the derivative of tan(x):

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

The derivatives of sin(x) and cos(x) are cos(x) and -sin(x), respectively.

d/dx(sin(x)) = cos(x)

d/dx(cos(x)) = -sin(x)

Plugging these values into the derivative formula, we get:

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

Simplifying:

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

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

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

Alternatively, we can express the derivative in terms of sin(x), using another trigonometric identity cos^2(x) = 1 – sin^2(x):

d tan(x) / dx = 1 / (1 – sin^2(x))

Therefore, the derivative of tan(x) with respect to x is 1 / (cos^2(x)) or 1 / (1 – sin^2(x)).

More Answers: