Understanding the Quotient Rule: Finding the Derivative of the Tangent Function

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 quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then the derivative is given by:

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

In this case, the tangent function can be written as f(x) = sin(x) / cos(x).

Using the quotient rule, we can find the derivative of f(x) with respect to x.

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

Simplifying the expression, we get:

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

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

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

Now, we can rewrite cos^2(x) as 1 – sin^2(x) using another trigonometric identity:

f'(x) = 1 / (1 – sin^2(x))

Finally, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute cos^2(x) with 1 – sin^2(x):

f'(x) = 1 / (1 – sin^2(x))
= 1 / cos^2(x)

Therefore, the derivative of the tangent function with respect to x is 1 / cos^2(x).

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